## 2 Materials and Methods
### Networked Model Predictive Formation Control
Assuming a Bernoulli characteristic of the network model in Equation (1) the control horizon \(N_{c}\) must satisfy the following criteria, so that within the time interval \(N_{c}T_{s}\) since the last successfully received packet at least one new control packet is transmitted and received successfully with probability \(p_{s}\):  
\[N_{c}\geq\frac{\log(1-p_{s})}{\log(p_{l})} \tag{6}\]  
The control horizon length \(N_{j}\) at time step \(j\) is regularly updated during run time so that the controller can react to changing network conditions. This is realized by adapting \(N_{j}\) whenever feedback from the actuator regarding its current state of the last updated control command buffer arrives at the controller node. Especially the number of already consumed/executed commands \(N_{con,j}\) from the stored command buffer at the actuator plays an important role. \(N_{j}\) is then smoothly updated via an asymmetric Exponentially-Weighted-Moving-Average (EWMA) filter using the following update rule:  
\[N_{j+1}= \begin{cases}\beta_{inc}\left(N_{min}+N_{con,j}+N_{th}\right)+\left( 1-\beta_{inc}\right)N_{j}&\text{if }N_{con,j}\geq N_{con,j-1}\\ \beta_{dec}\left(N_{min}+N_{con,j}+N_{th}\right)+\left(1-\beta_{dec}\right)N_{ j}&\text{if }N_{con,j}<N_{con,j-1}\end{cases}  \tag{7}\]  
By choosing \(0<\beta_{dec}<\beta_{inc}<1\) a faster adaption of \(N\) to increasing data loss on the network than to decreasing data loss events is realized, therefore following a more conservative adaption strategy. The threshold constant \(N_{th}\in\mathbb{N}\) is used to account for additional network uncertainty and increases the number of time steps until \(N_{con,j}\) reaches the \(N_{min}\) command reserve. This command reserve is chosen, so that in case of controller failure enough command reserve exists to facilitate a redistribution of the control task before the command buffer at the actuator runs out. The first control horizon length \(N_{0}\) is set to \(N_{0}=N_{min}+N_{th}+N_{pl,ps}\) with \(N_{p_{i},p_{s}}\) calculated according to Equation (6) using the expected network loss probability \(p_{l}\) and the desired probability \(p_{s}\) for at least one successful transfer from the controller before \(N_{con,j}\) reaches the \(N_{min}\) command reserve. Results of this adaption rule are shown in Section 3.2. To implement the MPC formulation given in Equation (5) on a digital controller, the problem needs to be reformulated as a discrete time problem with sample time \(T_{s}\). In addition, the system dynamics \(\mathbf{f}(\mathbf{x},\mathbf{u})\) are realized as discrete state space model of the relative satellite motion described in Equation (4) by assuming a sample-and-hold application of the control input \(\mathbf{u}_{i}\) to the system for the duration of the sample time \(T_{s}\). In addition we assume the prediction horizon length \(N\) and control horizon length \(N_{c}=N_{j}\) for time step \(j\) to be equal which does not impact stability as the Lyapunov stability constraint is independent of control and prediction horizon length [45]. This leads to the following discrete time MPC formulation with solution \(\mathbf{u}_{0;N}^{*}\):  
\[ \begin{split}\min_{\mathbf{u}_{0;N}\in\mathbb{U}}& \sum_{k=0}^{N}\tilde{\mathbf{x}}_{k}^{T}\mathbf{Q}\mathbf{x}_{k}+ \mathbf{u}_{k}^{T}\mathbf{R}\mathbf{u}_{k}\\ \text{s.t.}&\tilde{\mathbf{x}}_{k+1}=\mathbf{A}_{d} \,\tilde{\mathbf{x}}_{k}+\mathbf{B}_{d}\,\mathbf{u}_{k}\\ &\tilde{\mathbf{x}}_{k+1}=\mathbf{A}_{d}\,\tilde{\mathbf{x}}_{k} +\mathbf{B}_{d}\,\mathbf{h}(\tilde{\mathbf{x}}_{k})\\ &\tilde{\mathbf{x}}_{k}\in\mathbb{X}\\ &\tilde{\mathbf{x}}_{0}=\tilde{\mathbf{x}}_{0}\\ & V(\tilde{\mathbf{x}}_{k})\leq V(\tilde{\mathbf{x}}_{k}),\quad \forall k\in[0,...,N]\end{split}  \tag{8}\]  
With \(\mathbf{A}_{d}\) as system matrix and \(\mathbf{B}_{d}\) as control input matrix of the discrete time LTI model of the relative satellite motion with sampling time \(T_{s}\).