Point-Mass Satellite Moving in a Plane (1)

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