## 2 Materials and Methods
### Networked Model Predictive Formation Control  
The control method that we rely on for controlling a satellite in the formation is a MPC approach. MPC not only allows to tune the controlled system behavior for either optimal fuel usage or reference tracking depending on the application, but also can include complex input- and state constraints in the controller formulation at the same time. MPC has not only seen wide adoption in terrestrial applications like autonomous driving [40; 41; 42] but in the spacecraft domain as well [43]. The ever growing processing power of mobile processors and efficient SW implementations also allow the generation of online solutions to the MPC optimal control problem in real-time for many dynamic systems [44]. In the case of our networked closed-loop system we also need to guarantee stability in the event of successive packet loss in the sensor-node to controller-node and/or controller-node to actuator-node communication. For that we build upon a specific MPC formulation from [25] which is an extension of the networked MPC definition from [45]. Here, in order to cope with sensor to controller losses, the internal nominal system model is used to compensate for lost system state updates by predicting the missing states based on the last received measured ones. Losses in the controller to actuator direction are handled by sending the whole computed control trajectory of length \(N_{c}\) of the MPC at each sampling interval to the actuator and storing it there so that in the event of one or multiple actuation command losses the actuator-node executes the local stored control commands in an open-loop manner. One way to guarantee that the control trajectory \(u(t_{k}:t_{k+N_{c}})\) of one control horizon of length \(N_{c}\) leads to a stable system behavior in an open-loop condition, is to ensure that for a given Lyapunov function \(V(\mathbf{x})\) and a corresponding stable control law \(h(\mathbf{x})\) the MPC computed control input \(u(t_{k}:t_{k+N_{c}})\) leads to an equal or greater decrease of \(V(\mathbf{x})\) over the control horizon time as the control input generated by \(h(\mathbf{x})\) starting on the same initial conditions would lead to. The general networked MPC continuous time formulation is given below:  
\[ \begin{split}&\min_{\mathbf{u}}\int_{t_{k}}^{t_{k+N}}\bigl{[}|| \hat{\mathbf{x}}(\tau)||_{\mathbf{Q}}+||\mathbf{u}(\tau)||_{\mathbf{R}}\bigr{]} d\tau+F(\hat{\mathbf{x}}(t_{k+N}))\\ \text{s.t.}&\hat{\mathbf{x}}(t)=\mathbf{f}(\hat{ \mathbf{x}}(t),\mathbf{u}(t))\\ &\dot{\mathbf{x}}(t)=\mathbf{f}(\hat{\mathbf{x}}(t),h(\hat{ \mathbf{x}}(t_{k}+jT_{s})))\\ &\quad\quad\forall t\in[t_{k}+jT_{s},t_{k}+(j+1)T_{s}],\quad j\in[ 0,1,...,N-1]\\ &\mathbf{u}(t)\in\mathbb{U}\\ &\tilde{\mathbf{x}}(t)\in\mathbb{X}\\ &\hat{\mathbf{x}}(t_{k})=\tilde{\mathbf{x}}(t_{k})\\ & V(\tilde{\mathbf{x}}(t))\leq V(\hat{\mathbf{x}}(t)),\quad \forall t\in[t_{k},t_{k}+N_{c}T_{s}]\end{split}  \tag{5}\]  
With \(N\geq N_{c}\) as the prediction horizon length, \(N_{c}\) as the control horizon length, \(T_{s}\) as the control sample time, \(\tilde{\mathbf{x}}\) as the propagated state subject to the MPC control inputs, \(\hat{\mathbf{x}}\) as the propagated state subject to the Lyapunov stable controller inputs in a sample-and-hold fashion, \(\mathbf{f}(\mathbf{x},\mathbf{u})\) as the system dynamics, \(\mathbb{U}\) as the allowed control input set and \(\mathbb{X}\) as the admissible state set. The quadratic cost on the state is given by \(||\mathbf{x}(t)||_{\mathbf{Q}}=\mathbf{x}^{T}(t)\mathbf{Q}\mathbf{x}(t)\) with \(\mathbf{Q}=\mathbf{Q}^{T}\) and \(\mathbf{Q}>0\). The quadratic cost on the control input is given by \(||\mathbf{u}(t)||_{\mathbf{R}}=\mathbf{u}^{T}(t)\mathbf{R}\mathbf{u}(t)\) with \(\mathbf{R}=\mathbf{R}^{T}\) and \(\mathbf{R}>0\). The final state cost penalty \(F(\mathbf{x})\) that is often used to guarantee stability of the MPC formulation will be omitted as the Lyapunov constraint imposed on the state trajectory in the MPC formulation already serves as a stability guarantee.