SI&DA - Lecture 18 'Optimal Model Choice'

$\phi$ : regressor → In case of ARX models the regressor does not depend on the unknown “parameter vector” $\theta$ → In case of the other models, it does depend on it.

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Where:

• $N$ : max value of $t$, $t\in \left[t_0,N\right]$
• $Z$ : Is the data set
• So $Z^N$ is the Data Set up to time $N$.

(In this case it’s a least square function) The filter $L(z)$ applied to the prediction error $\epsilon$ is the same as filtering both output ($y(t)$) and input ($u(t)$). We can see $L(z)H(z{)}^{-1}$ as a single filter.

A reason to design $L(z)$ could be if we are not interested in high frequency error, for example our system could have no response to high frequency input, then we could design $L(z)$ as a low pass filter, or another more general, example could be that we are just interested in a specific band of the system (pass-band filter).

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We can create an ARX cost function as: Then we will have that: Also remember that $R(N)\in {\mathbb{R}}^{d\times d}$

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Where in the last formula we can see:

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