A controlled Van der Pol system can be described by the following equation

$$\begin{align*} \dot{x}_{1} &= x_{2} \\ \dot{x}_{2} &= -x_{1}+\epsilon (1-x_{1}^{2})x_{2}+u, \epsilon > 0 \\ y &= x_{1} \end{align*}$$

consider a vibration system with nonlinear springs shown in the attached image. The normal form of the system is given by

$$\begin{align*} \dot\xi_{1}^{1} &= \xi_{2}^{1} \\ \dot{\xi}_{2}^{1}&=-\frac{k_1}{m_{1}}(\xi_{1}^{1}-\xi_{1}^{2})-{\bar{k} _{1}\over m_{1}}(\xi_{1}^{1}-\xi_{1}^{2})^{3}-{c_1\over m_{1}}(\xi_{2}^{1}-\xi_{2}^{2})+{u_1\over m_{1}} \\ \dot{\xi}_{1}^{2} &= \xi_{2}^{2} \\ \dot\xi_{2}^{2}&=\frac{k_1}{m_{2}}(\xi_{1}^{1}-\xi_{1}^{2})+{\bar{k} _{1}\over m_{2}}(\xi_{1}^{1}-\xi_{1}^{2})^{3}+{c_1\over m_{2}}(\xi_{2}^{1}-\xi_{2}^{2}) \\ &-{k_2\over m_{2}}(\xi_{1}^{1}-\xi_{1}^{2})-{k_2\over m_{2}}(\xi_{1}^{1}-\xi_{1}^{2})^{3}-{c_2\over m_{2}}(\xi_{2}^{1}-\xi_{2}^{2})+{u_2\over m_{2}} \\ \dot{\eta}_{1} &= \eta_{2} \\ \dot{\eta}_{2} &= {k_2\over m_{3}}(\xi_{1}^{2}-\eta_{1})+{\bar{k} _{2}\over m_{3}}(\xi_{1}^{2}-\eta_{1})^{3}+{c_2\over m_{3}}(\xi_{2}^{2}-\eta_{2}) \\ &-{k_3\over m_{3}}\eta_{1}-{\bar{k} _{3}\over m_{3}}\eta_{1}^{3}-{c_3\over m_{3}}\eta_{2} \\ y_{1} &=\xi_{1}^{1} \\ y_{2} &=\xi_{1}^{2}, \end{align*}$$

where parameters $m_i, k_i$ and $\bar{k} (i=1,2,3)$ are positive constants. The system has the relative degrees $r_1 = 2, r_2 = 2$ and the following zero dynamics.

$\begin{align*} \dot{\eta}_{1} &=\eta_{2} \\ \dot{\eta}_{2} &=-\frac{k_2}{m_3}\eta_{1}-\frac{\bar{k}_2}{m_3}\eta_1^3+\frac{c_2}{m_3}(\xi_2^2-\eta_2) \\ &-\frac{k_3}{m_3}\eta_1-\frac{\bar{k}_3}{m_3}\eta_1^3-\frac{c_3}{m_3}\eta_2 \end{align*}$$

Consider nonlinear van der Pol oscillators coupled via a bath. The normal form of the system is expressed by

$$\begin{align*} \dot{\xi}_{1}^{1}&=\xi_{2}^{1} \\ \dot{\xi}_{2}^{1}&=-\xi_{1}^{1}+\epsilon\{1-(\xi_{1}^{1})^{2}\}\xi_{2}^{1}+k(\eta_{1}-\xi_{1}^{1})+u_{1} \\ \dot{\xi}_{1}^{2}&=\xi_{2}^{2} \\ \dot{\xi}_{2}^{2}&=-\xi_{1}^{2}+\epsilon\{1-(\xi_{1}^{2})^{2}\}\xi_{2}^{2}+k(\eta_{1}-\xi_{1}^{2})+u_{2} \\ \dot{\eta}_{1}&=k(\xi_{1}^{1}-\eta_{1})+k(\xi_{1}^{2}-\eta_{1}) \\ y_{1}&=\xi_{1}^{1} \\ y_{2}&=\xi_{1}^{2}, \end{align*}$$

where parameters $\epsilon$ and $k$ are positive constants. The system has the relative degrees $r_1 = 2, r_2 = 2$ and the following zero dynamics:

$ \dot{\eta}_{1}=-2k\eta_{1}.$

The inverted pendulum on cart to be controlled is shown in the attached image. Its structure consists of a cart and pendulum where the pendulum is hinged to the cart via a pivot and only the cart is actuated, where:

Lagrange's equations are applied with respect to $\theta$ and $x$ coordinates. The nonlinear state space model of inverted pendulum on cart system can then be obtained as:

$$\begin{align*} \dot{x}_1 &= x_2 \\ \dot{x}_2 &= \dfrac{mg \sin{x_1}+ \dfrac{ m\cos{x_1}(u-f\dot{x})-m^2lx_2^2\cos{x_1}\sin{x_1}}{(M + m)}}{\dfrac{4}{3}ml - \dfrac{m^2l\cos^2{x_1}}{(M+m)}} \\ \dot{x}_3 &= x_4 \\ \dot{x}_4 &= \dfrac{u - f x_1 + \dfrac{3}{4}mg\cos{x_1}\sin{x_1} - mlx_2^2\sin{x_1}}{(M+m) - \dfrac{3}{4}m\cos^2{x_1}}, \end{align*}$$

where the state variables are consequently assigned as $x_1 = \theta$, $x_2 = \dot{\theta}$, $x_3 = x$ an $x_4=\dot{x}$, and where the input $u$ is the applied force $F$.

The nonliear model for elevator subsystem is as follows:

$$ f(x,u) = \begin{bmatrix} x_2\\ \dfrac{1}{I}(-\tau_g \sin{x_1} +k_{gyro}(u_1x_6\cos{x_1})- B_{\psi} x_2 + a_1 x_3 ^2 + b_1 x_3 )\\ x_4 \\ \dfrac{1}{T_1^2}(u_1 -x_3-2T_1x_4)\\ x_6 \\ \dfrac{1}{I_{\phi}} \left(-B_{\phi}x_6 - \left[K_r \dfrac{T_{or}}{T_{pr}}u_1 + x_9\right] + a_2x_7^2 + b_2x_7\right) \\ x_8 \\ \dfrac{1}{T_2^2}(u_2 - x_7 -2T_2x_8)\\ \dfrac{1}{T_{pr}}\left[K_r \left(1- \dfrac{T_{or}}{T_{pr}}\right)u_1 - x_9\right]\\ \end{bmatrix},$$

where $x_1$ is the elevator angle, $x_5$ is azimuth angle, and $g(x)=\begin{pmatrix}x_1k_{\psi} + y_{\psi \circ} \\ x_2k_{\phi} + y_{\phi \circ }\end{pmatrix}$