Recursive Generalized Least Squares RGLS

In this case we assume again the disturbance term, e , is not white noise, rather it is related to 4n through the following transfer function (noise filter) 

The above equations suggest that the unknown parameters in polynomials A( ) and B() can be estimated with RLS with the transformed variables yn and un k. Having polynomials A( ) and B(-) we can go back to Equation 13.1 and obtain an estimate of the error term, e , as 

The above equation cannot be used directly for RLS estimation. Instead of the true error terms, e , we must use the estimated values from Equation 13.35. Therefore, the recursive generalized least squares (RGLS) algorithm can be implemented as a two-step estimation procedure  

Apply RLS on equation A(z )yn = B(z )un k + based on information up to time t . Namely, obtain the updated value for the parameter vector (i.e. the coefficients of the polynomials A and 

C(z l) which are used for the computation of the transformed variables yn and un )c at the next sampling interval. 

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