# Modeling and Control of Satellite Formations: A Survey
## 2 Multiple-Input-Multiple-Output Formation Architecture
For example, [54] considers the dynamics of \(N\) agents given by the linear difference state equations  
\[x[k+1]=Ax[k]+Bu[k],\quad y_{i}[k]=Cx[k]+n_{i}[k],i=1,\ldots,N, \tag{16}\]  
where \(x[k]\in\mathbb{R}^{n}\) is the system state vector, \(u[k]\in\mathbb{R}^{mN}\) is vector control, \(y_{i}[k]\) are the measured outputs of agents, \(n_{i}[k]\)--measurement errors, \(i=1,\ldots,N\). It is assumed that each vehicle has the same number \(m\) of executive devices, and also that each agent with the number \(i\) has a local controller that generates the corresponding control signal \(u_{i}[k]\). Matrices \(Q_{i}\in\mathbb{R}^{m\times mN}\) are introduced such that \(u_{i}[k]=Q_{i}u[k]\) and projections \(\Pi_{i}=Q_{i}^{T}Q_{i}\), which together form the identity matrix \(\mathbf{I}_{mN}=\sum_{i=1}^{N}\Pi_{i}\). The \((A,C)\) pair is considered observable. A stabilizing feedback \(u[k]=Kx[k]\) is used such that the matrix of equations of state of the entire closed-loop system \(A_{\mathrm{clp}}=A+BK\) is asymptotically stable (its spectrum lies inside a circle of unit radius). This leads to \(N\) dynamic controllers according to the state estimates \(u_{i}[k]=Q_{i}K\hat{x}_{i}[k]\), in which the \(\hat{x}_{i}[k]\) estimates are generated by the observers  
\[\hat{x}_{i}[k+1]=A\hat{x}_{i}[k]+BK\hat{x}_{i}[k]+L\big{(}C\hat{x}_{i}[k]-y_{i} [k]\big{)}, \tag{17}\]  
where \(L\) is the observer's feedback matrix to be chosen in the synthesis. Each control signal \(u_{i}[k]\) is generated based on a local estimate of the state of the entire system \(\hat{x}_{i}[k]\). This approach uses an estimate of the actuator input applied by each of the other vehicles and will generally not be correct for non-local components. The incompleteness of information about the input data of the object leads to the fact that the dynamics of the estimation error become bounded. Further, [54] uses the well-known method of communication specification based on graph theory [56, 57]. It was found that communication between vehicle observers can be used to eliminate or mitigate the consequences of the interconnecteddynamics of estimation errors. The communication structure and analysis presented in [54] provide a means of designing a communication network that can determine the dynamics of interconnected errors. It is shown that in this structure the minimum number of communication channels required to eliminate these dynamics grows only linearly with the size of the formation. Another result is that in terms of network robustness and resiliency, \"receivers are much more valuable than transmitters\". A similar problem is considered in [58], where the exchange of navigation data between two satellites is considered, aimed at reducing the load of the inter-satellite communication channel, taking into account the dynamics of the relative motion of the satellites and possible erasures in the navigation data of the channel. For this, an adaptive binary coding/decoding procedure is used to transmit navigation data between satellites and a closed-loop control law that regulates the relative motion of the satellites. Using the calculus of variations, [PERSON] [24] deduced the necessary conditions for the optimal fuel consumption problem of multi-impulse rendezvous, known as the primer vector theory (see also [59]). In [60], a numerical method using [PERSON]'s necessary conditions is proposed. [PERSON]'s theory is applied in [61] to the task of rendezvous of the Apollo spacecraft. Research in this direction was developed in [62; 63; 64; 65; 66]. [PERSON] et al. [63] used basis vector theory to find optimal time-fixed multi-impulse coplanar and bounded class non-coplanar rendezvous solutions.