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

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. $\sigma = 0.1$.

$$\begin{align*} y(t) &= y(t-1) + u(t-1) + 1.3u(t-2) + 0.3u(t-1)y(t-1) \\ &+0.5u(t-2)y(t-2)+e(t)/\Delta, \end{align*}$$

where $e(t)$ is normal school white noise signal with covariance 0.1.

The ball and plate system is a system, where a metal ball stays on a rigid square plate with each side length of 1m. The slope of the plate can be manipulated by two perpendicularly installed step motors, so that the tilting of the plate will make the ball moving.

$$\begin{align*} \begin{bmatrix} \dot{x}_1 \\ \dot{x}_2\\ \dot{x}_3\\ \dot{x}_4\\ \dot{x}_5\\ \dot{x}_6\\ \dot{x}_7\\ \dot{x}_8\\ \end{bmatrix} &= \begin{bmatrix} x_2 \\ B(x_1x_4^2 + x_4x_5x_8 - g \sin x_3)\\ x_4\\ 0\\ x_6\\ B(x_5x_8^2 + x_1x_4x_8 - g \sin x_7)\\ x_8\\ 0\\ \end{bmatrix} + \begin{bmatrix} 0 & 0\\ 0 & 0\\ 0 & 0\\ 1 & 0\\ 0 & 0\\ 0 & 0\\ 0 & 0\\ 0 & 1\\ \end{bmatrix} \begin{bmatrix} u_x\\ u_y\\ \end{bmatrix}, \\ Y &= h(X) = (x_1,x_5)^{\mathrm T}, \end{align*}$$

where $B= m/(m + J/R^2)$ and $X = (x_1; x_2; x_3; x_4; x_5; x_6; x_7; x_8)^{\mathrm T} = (x; \dot{x}; \theta_x; \dot{\theta}_x; y; \dot{y}; \theta_y; \dot{\theta}_y)^{\mathrm T}$

Parameters are presented in the table below.