A nonlinear ARX (NARX) system

Model description: 

$$\begin{align*} y(t)&=\begin{bmatrix}0.1 & -0.1 & 0.25 & 0.5 \end{bmatrix} \varphi (t) +\\ &+ \frac{L}{2}(\|\varphi (t)\|^2 - 2(max\{\|\varphi(t)\|^2, 1\} -1) +\\ &+ 2(max\{\|\varphi(t)\|^2, 2\} - 2) - (max\{\|\varphi(t)\|^2, 3\} -3)) + e(t), \end{align*} $$

where

$\varphi(t)=\begin{bmatrix} y(t-1) & y(t-2) & u(t-1) & u(t-2)\end{bmatrix}^{\mathrm T}$

$L = 0.1$

$e(t) \in N(0, 0.01)$, i.e. $\sigma = 0.1$.

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Publication details: 

TitleNonlinear system identification via direct weight optimization
Publication TypeJournal Article
AuthorsRoll, Jacob, Nazin Alexander, and Ljung Lennart