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:

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

$$\begin{align*} y_{1}(k+1) &=0.9y_{1}(k)-0.3y_{1}(k-1)/[1+y_{2}^{2}(k-1)]+0.7u_{1}(k) \\ &+0.1y_{1}^{2}(k-1)y_{2}^{2}(k)+0.3\sin(u_{1}(k-1))-0.7u_{2}(k)+0.6u_{2}(k-1) \\ y_{2}(k+1) &=-0.1y_{2}(k-1)+0.3y_{1}(k-1)y_{2}(k)+0.8\sin(u_{1}(k)) \\ &+0.1u_{1}(k-1)+0.9u_{2}(k)+0.2u_{2}(k-1)+0.1u_{2}^{2}(k-1) \end{align*}$$

The RTNN model is described bythe following equations:

$$\begin{align*} X(k+1) &= JX(k) + BU(k)\\ Z(k) &= S[X(k)]\\ Y(k) &= S[CZ(k)]\\ J &\doteq \mathrm{blockdiag}(J_i); |J_i| <1, \end{align*}$$

here $X(\cdot)$ is a $n$-state vector of the RTTN; $U(\cdot)$ is a $m$-input vector; $Y(\cdot)$ is a $l$-output vector; $Z(\cdot)$ is an auxiliary vector variable with $l$ dimension; $S(\cdot)$ is a vector-valued smooth activation function (sigmoid, $tanh$, saturation) with appropriate dimensions; $J$ is a weigh-state block-diagonal matrix with $(1 \times 1)$ and $(2 \times 2)$ blocks; $J_i$ is an $i-th$ block of $J$ and $|J_i|<1$ is a stability condition.

Consider a truck-trailer system depicted in the attached image. Its dynamics is described by

$$\begin{align*} x_{1}(t+1) &=\left(1-\frac{vT}{L}\right)x_{1}(t)+\frac{vT}{l}u(t) \\ x_{2}(t+1) &=\frac{vT}{L}x_1(t)+x_{2}(t) \\ x_{3}(t+1) &=x_{3}(t)+vT\sin\left(\frac{vT}{2L}x_{1}(t)+x_{2}(t)\right)x_{1}(t), \end{align*}$$

where $x_1(t)$ : angle difference between truck and trailer. $x_2(t)$ : angle of trailer. $x_3(t)$ : vertical position of rear of trailer, $u(t)$ : steering angle, $T$ : sampling time. In this example, the parameters are $T=2.0s$, $l=2.8m$, $L=5.5m$, $v=-1.0m/s$.