This is a more complex model of the system in Point-Mass Satellite Moving in a Plane (1).

$$\begin{align*} \dot{y}_{1} &=x_{1,2}+\psi(y_{2}) \\ \dot{x}_{1,2} &=u_{1} \\ \dot{y}_{2} &=u_{2}. \end{align*}$$

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 $x^{\mathrm T}=[i_a, i_b, i_c, i_d]^{\mathrm T}.$

Using

$z_{i+1,k+1}=y_{k+1}^{(i)}\approx y_{k}^{(i)}+Ty_{k}^{(i+1)}+\frac{T^2}{2}y_{k}^{(i+2)}+\cdots+\frac{T^{r-i}}{(r-i)!}y_{k}^{(r)}+\frac{T^{r-i+1}}{(r-i+1)!}y_{k}^{(r +1)}$

for $i=0, \cdots,r-1$ the controlled Van der Pol system from Controlled Van der Pol system (1) can be rewritten as:

$$\begin{align*} x_{1,k+1} &=x_{1,k}+Tx_{2,k}+\frac{T^2}{2}[-cx_{2,k}-d\sin x_{1,k}+u_{1,k}] \\ &+\frac{T^{2}}{3!}[-c(-cx_{2,k}-d\sin x_{1,k}+u_{1,k})-dx_{2,k}\cos x_{1,k} \\ &\times\{-x_{1,k}+\epsilon(1-x_{1,k}^2)x_{2,k}+u_k] \\ x_{2,k+1} &=x_{2,k}+T[-cx_{2,k}-d\sin x_{1,k}+u_{1,k}] \\ &+\frac{T^2}{2}[-cx_{2,k}-d\sin x_{1,k}+u_{1,k}-dx_{2,k}\cos x_{1,k} \\ &\times\{-x_{1,k}+\epsilon(1-x_{1,k}^2)x_{2,k}+u_k] \\ y_{k} &=x_{1,k} \end{align*}$$

Consider the pendulum system with the relative degree two

$$\begin{align*} \dot{x}_{1} &= x_{2}\\ \dot{x}_{2} &= -cx_{2}-d\sin x_{1}+u \\ y &= x_{1} \end{align*}$$

with $c > 0$.