The time-invariant bilinear system is given by

$$Y(t) = 1.5X(t) + 1.2X(t-1) - 0.2X(t-2) + 0.7X(t-1)Y(t-1) - 0.1X(t-2)Y(t-2) + \epsilon(t),$$

where $A=0, \alpha=0, B=\begin{bmatrix}1.5 &1.2 &-0.2\end{bmatrix}, C = \begin{bmatrix}0.7 &0 &-0.1\end{bmatrix}$. Note that $\Theta = \begin{bmatrix}B & C\end{bmatrix}^{\mathrm T}.$

Consider the following bilinear descriptor system:

$$\begin{pmatrix} 1 & -1\\ 0 & 0 \end{pmatrix}x_{k+1}=\begin{pmatrix} -0.5 & 1\\ -1 & 0 \end{pmatrix}x_{k}+\begin{pmatrix} 0.5 & 0.25\\ -1 & 0.5 \end{pmatrix}x_{k}u_{k}.$$

Consider a linear system represented by the transfer function

$$G(s)=\dfrac{c}{s(s+a)}$$

where $a$ and $c>0$ are unknowns constants, and the reference model

$$G_m(s)=\dfrac{\omega^2}{s^2 + 2\zeta\omega s + \omega^2}.$$

The system

$$\begin{align*} \dot{x}_1 & = x_4^2 + x_3^3 + u_1 + au_2 \\ \dot{x}_2 & = x_3 \\ \dot{x}_3 & = \sin{x_4}+\cos{x_1}+bu_1 + u_2 \\ \dot{x}_4 & = -x_4 \\ y_1 &= x_1 \\ y_2 &=x_2. \end{align*}$$

The system

$$\begin{align*} \dot{x}_1 & = x_2 \\ \dot{x}_2 & = x_3^2 + x_4 + u_1 + au_2 \\ \dot{x}_3 & = x_4 + bu_1 + u_2 \\ \dot{x}_4 & = -x_4 \\ y_1 &= x_1 \\ y_2 &=x_3 \end{align*}$$

with $ab \neq 1$, has a well-defined vector relative degree (2, 1) and a nonsingular decoupling matrix $\begin{bmatrix} 1 & a \\ b & 1 \end{bmatrix}$.