The system consists of a DC water pump feeding a conical flask which in turn feeds a square tank, giving the system second-order dynamics. The controllable input is the voltage to the pump motor and the system output is the height of the water in the conical flask. The aim, under simulation conditions, is for the water height to follow some demand signal. The plant model was identified as

$$\begin{align*}z(t) &=0.9722z(t-1)+0.3578u(t-1)-0.1295u(t-2)-\\ &-0.3103z(t-1)u(t-1)-0.04228z^6(t-2)+0.1663z(t-2)u(t-2)+\\ &+0.2573z(t-2)e(t-1)-0.03259z^2(t-1)z(t-2) - 0.3513z^2(t-1)u(t-2)+\\ &+0.3084z(t-1)z(t-2)u(t-2)+0.2939z^2(t-2)e(t-1)+\\ &+0.1087z(t-2)u(t-1)u(t-2)+0.4770z(t-2)u(t-1)e(t-1)+\\ &+0.6389u^2(t-2)e(t-1)+e(t), \end{align*}$$

where $e(t)$ is a noise.

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

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 system has two inputs and two outputs and is described by the following set of equations:

$$\begin{align*} y_1(k)&=0.21y_1(k-1)-0.12y_2(k-2)+0.3y_1(k-1)u_2(k-1) \\ &-1.6u_2(k-1)+1.2u_1(k-1)\\ y_2(k)&=0.25y_2(k-1)-0.1y_1(k-2)-0.2y_2(k-1)u_1(k-1) \\ &-2.6u_1(k-1)-1.2u_2(k-1). \end{align*}$$