The point-mass satellite moving in a plane, subject to an inverse-square law force field

$$\begin{align*} \dot{r} &=v \\ \dot{v} &=r\omega^{2}-a_{1}\frac{1}{r^2}+a_{2}u_{1} \\ \dot{\phi} &=\omega \\ \dot{\omega} &=-2\frac{v\omega}{r}+a_{2}\frac{1}{r}u_2, \end{align*}$$

where the states are: the satellite's radial position $r < 0$, its radial velocity $v$, its angular position $\phi$ and its angular velocity $\omega$. The satellite is equipped with a radial thruster $u_1$ and a tangential thruster $u_2$, and the known constant model parameters $a_1$ and $a_2$ are related to the strength of the force field and the mass of the satellite. If the outputs are taken as $y_1=\sin{\phi}$ and $y_2=\cos{\phi}$, then the transformation $z_{1,1}=r\sin{\phi}$, $z_{1,2}=v\sin{\phi}+r\omega\sin{\phi}$, $z_{2,1} = r\cos{\phi}$ and $z_{2,2}=v\cos{\phi} - r\omega\cos{\phi}$, brings the system into the NOF

$$\begin{align*} \dot{z}_{1,1} &=z_{1,2} \\ \dot{z}_{1,2} &=-a_{1}\frac{y_1}{\left(y_{1}^{2}+y_{2}^{2}\right)^{3/2}}+\bar{u}_{1} \\ \dot{z}_{2,1} &=z_{2,2} \\ \dot{z}_{2,2} &=-a_{1}\frac{y_2}{\left(y_{1}^{2}+y_{2}^{2}\right)^{3/2}}+\bar{u}_{2}, \end{align*}$$

where $\bar{u}_1 = a_2 ((u_1y_1 + u_2y_2)/(y_1^2 + y_2^2)^{1/2})$ and $\bar{u}_2 = a_2 ((u_1y_2 + u_2y_1)/(y_1^2 + y_2^2)^{1/2})$.

The equations describing the stepped motor in the attached image are given as follows:

$$\begin{align*} \dot{x}_1 &= -K_1x_1 + K_2x_3\sin{(K_5x_4)} + u_1 \\ \dot{x}_2 &= -K_1x_2 + K_2x_3\sin{(K_5x_4)} + u_2 \\ \dot{x}_3 &= -K_3x_1\sin{(K_5x_4)} + K_3x_2\cos{(K_5x_4)} - K_4x_3 - K_6\sin{(4K_5x_4)} - \tau_L/J \\ \dot{x}_4 &= x_3, \end{align*}$$

where $K_1 = R/L$, $K_2 = K_m/L$, $K_3=K_m/J$, $K_4 = B/J$, $K_5 = N_r$, $K_6 = K_D/J$, $K_5 = N_r$, $K_6 = K_D/J$, $u_1=v_a/L$.

The term $K_D\sin{(4N_r\theta)}$ represents the detent torque due to the permanent rotor magnet interacting with the magnetic materia of the stator poles. $K_D$ is typically 5% to 10% of the value of $K_mi_0$, where $i_0$ is the rated current.

The state variables $x_1, x_2, x_3$ and $x_4$ are assigned by $x^{\mathrm T}=[i_a, i_b, i_c, i_d]^{\mathrm T}.$

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*}$$

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 $g(x)=\begin{pmatrix}x_1k_{\psi} + y_{\psi \circ} \\ x_2k_{\phi} + y_{\phi \circ }\end{pmatrix}$