## 2 Materials and Methods
### Satellite Formation-Dynamic Model  
The primary control objective that needs to be maintained in the presence controller loss is formation control of two or more satellites. In the following paragraph the dynamic model that describes the motion of the satellites in the formation is introduced. The formation description utilized in this work is a so-called leader-follower approach, with one designated leader satellite \(S_{0}\) also called chief and one or more follower satellites \(S_{[1,2,\ldots]}\), also called deputies. The chief satellite is uncontrolled within the formation and can be a real satellite or virtual reference point, whereas the deputy satellites are controlled to achieve a certain relative position and movement with respect to the chief. The motion of each individual satellite can be described in an Earth-Centered Inertial (ECI) frame by the position vector \(\mathbf{r_{i}}\) from the center of the Earth to the position of the satellite along its orbit and the velocity vector \(\mathbf{v_{i}}=\dot{\mathbf{r_{i}}}\). The motion of a single satellite \(S_{i}\) in Earth's gravitational potential with respect to the ECI reference frame can be described as follows [38]:  
\[\ddot{\mathbf{r_{i}}}=-\
abla U_{g}+\mathbf{F_{d,i}}+\mathbf{F_{u,i}} \tag{2}\] \[\text{with}\quad\
abla U_{g}=\frac{\mu}{r_{i}^{3}}\mathbf{r_{i}}\]  
where \(U_{g}\) is describing the Earth gravitational potential field assuming a perfect homogeneous mass distribution of Earth. \(\mathbf{F_{d,i}}\) contains all disturbance forces per unit mass acting on satellite \(S_{i}\), \(\mathbf{F_{u,i}}\) is the controlled thrust force per unit mass and \(\mu\) is the gravitational constant of the Earth. The relative movement of the deputy satellites with respect to the chief \(S_{0}\) can be described in a chief centered coordinate frame, centered at the position of \(S_{0}\). This so-called Local-Vertical, Local-Horizontal (LVLH) frame is depicted in Figure 5 and its unit vectors are defined as follows: \(\hat{X}\) is radially outwards, \(\hat{Y}\) is in along-track/satellite velocity direction and \(\bar{Z}\) is perpendicular to the orbital plane. With \(\boldsymbol{\rho_{i}}=\mathbf{r_{i}}-\mathbf{r_{0}}=[x_{i},y_{i},z_{i}]^{T}\) as the relative position of the i-th deputy satellite and \(\mathbf{r_{0}}\) as the osculating ECI position of the chief. Figure 5: Local-Vertical, Local-Horizontal coordinate frameThe model of the relative dynamics that we use for the i-th satellite in the LVLH frame are the well known Hill-Clohessy-Wiltshire (HCW) Equations [39] given by:  
\[\dot{x_{i}}-2 ny_{i}-3n^{2}x_{i} =u_{i,x}\] \[\dot{y_{i}}+2n\dot{x_{i}} =u_{i,y} \tag{3}\] \[\dot{z_{i}}+n^{2}z_{i} =u_{i,z}\]  
with mean motion \(n=\sqrt{\mu/r_{0}^{3}}\) and \(\mathbf{u}_{i}=[u_{i,x},u_{i,y},u_{i,z}]^{T}\) as control input to the satellite. They provide a Linear Time-Invariant (LTI) description of the non-linear relative satellite motion under several assumptions. First we assume a circular orbit of the real or virtual reference satellite \(S_{0}\) which results in a constant mean motion \(n\) and norm of \(\mathbf{r_{0}}\). As second assumption we consider a scenario with close proximity of the satellites so that inter-satellite node communication can be realized. This allows to neglect differential disturbance forces and other higher order un-modelled terms of the relative-distance satellite dynamics in the employed linear HCW formation model. These model simplifications pose no loss of generality of the proposed approach in this work as more complex non-linear relative-dynamic models could be incorporated in the same way by choosing an appropriate solver of the networked Optimal Control Problem (OCP). Indeed the GRAMPC solver utilized in this work is capable of solving non-linear OCP problems as well, which was demonstrated for a networked OCP with a non-linear formation dynamics model in [27].