Cross-correlation and Autocorrelation

The modeling error at every time instant can be written as  

The model parameter vector can now be computed by minimizing J= (l/2) this can be achieved by setting dJ /dd = 0, from which  

We have discussed the model stmcture and model identification procedure. If data are available, we would essentially be able to identify the model parameters. There are a few pitfalls, however. The data that are or will be collected should be appropriate data for model identificatioa This means that the data set should contain sufficient dynamic and static information, such that the model can be identified and reflects the tine process behavior. 

Often normal process operating data are not fit for dynamic model identification. It just does not contain sufficient information, resulting in a poor model. The process input is therefore usually perturbed, for example by using a Pseudo-Random Binary Sequence (PRBS) or any other form of perturbation. The PRBS signal provides a sequence of upward and downward steps as shown in Fig. 24.1. 

Once a proper PRBS signal has been identified and process data has been collected, one should determine the cross correlation of the process input-output signals and the autocorrelation of the process output. 

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It should be noted that the inverse Fourier transforms of crosspower and autopower spectra are the time domain cross correlation and autocorrelation functions, respectively. Thus, another entry into FT-FAM is to record time domain cross and autocorrelation functions, Fourier transform these, and divide to yield the cell admittance or impedance. Such procedures have been applied.33,34... 

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