Predictability of Process Models

Since the models obtained in system control are often used to predict or forecast futiue behaviour of a system, it is important to develop an understanding of the predictive properties of a model. This can be formalised by considering the w-step ahead predictor for a process, + which predicts the process value w-steps ahead given all the values of the process up until the current point t and the input up until the point t- -r. Let the prediction error be defined as 

6 Modelling Dynamic Processes Using System Identification Methods 

Theorem 6.1 (z-step ahead linear predictor) The m-step ahead linear predictor, 

Proof The proof of this theorem can be obtained by solving Eq. (6.12) for the case of a linear predictor given by Eq. (6.13) and the model by Eq. (6.2). Many of the steps will be similar to those used to obtain the time series predictor in Sect. 5.4.4. Substituting Eqs. (6.13) and (6.2) into Eq. (6.12) gives 

In order to separate the available error values from those which are not, the impulse response model of the disturbance transfer function will now be split into two components a term containing all the terms up to, but excluding, the rth impulse term (unavailable future component) and a term containing all the remaining terms (available past component). It can be noted that the last term can be rewritten as a difference between the original disturbance transfer function and the unavailable future component, that is, 

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