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 $\begin{bmatrix} 1 & a \\ b & 1 \end{bmatrix}$.

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 $T=2.0s$, $l=2.8m$, $L=5.5m$, $v=-1.0m/s$.

The behavior of phytoplankton cells in a continuous reactor is usually described by the Droop model. Cell growth is limited by a nutrient with concentration $S$. The biomass has a concentration $N$ and $Q$ represents the cell quota of assimilated nutrient, expressed as the amount of intracellular nutrient per biomass unit. The dilution rate $D$ corresponds to the flow rate of renewal medium over the volume of the reactor, and $D$ is the input of the system.

We denote $D = D_0 + u$, and the system fits

$$\sum_D \begin{cases} \dot{x}_i = f(x) + ug(x)\\ y=h(x_1) \end{cases}$$

with

$f(x)=\begin{pmatrix} a_2\left(1-\dfrac{1}{x_2}\right)x_1 - D_0x_1\\ a_3\dfrac{x_3}{a_1+x_3} - a_2(x_2 - 1)\\ D_0(1-x_3)-\dfrac{x_1x_3}{a_1+x_3} \end{pmatrix}$

$g(x)=\begin{pmatrix} -x_1\\ 0\\ 1-x_3 \end{pmatrix}$, and $h(x_1)=x_1$, where

$ x_1 = (\rho_m N/S_i);\\ x_2 = (Q/K_Q);\\ x_3 = (S/S_i);\\ a_1 = (K_{\rho}/S_i);\\ a_2 = \mu_m;\\ a_3 = (\rho_m/K_Q). $