# Modeling and Control of Satellite Formations: A Survey
## 4 Leader-Follower Formation Architecture  
Fulfillment of the laws of linear control for the satellites formation in the presence of gravitational disturbances is considered by [PERSON] [110]. The use of a linear quadratic controller that minimizes the error between the actual and the required relative motion of the satellite is evaluated for the ability to maintain a specific geometry of the formation in the presence of such a gravitational disturbance as the uneven gravity of the Earth. The required formation geometry is based on solving linear equations of relative motion without disturbances. In particular, a group of satellites is selected whose projected motion onto the tangential plane of the Earth is a circle with a radius of one kilometer. It is shown that linear control laws support the formation within the error in the presence of gravitational disturbances. In addition, the simulation provides an estimate that takes into account the required fuel consumption to maneuver such a formation, thereby providing studies based on reducing costs while moving along the desired trajectories and according to the selected control strategies. In [111], the formation control problem is considered in the following general setting. A complex system consisting of many subsystems (mobile agents) is described by the following equations  
\[\dot{x}_{i}=f_{i}(x_{i},u_{i},r_{i}),\quad y_{i}=h_{i}(x_{i}),\quad i=1,\ldots,k \tag{43}\]  
where \(k\) is the number of subsystems; \(x_{i}\in\mathbb{R}^{n_{i}}\) is the state vector of the \(i\)-th subsystem; \(r_{i}\) express interrelations between subsystems and are defined as functions of \((x_{i},u_{j})\) for \(i\
eq j\); \(u_{i}\in\mathbb{R}^{m_{i}}\) denotes the control input action of the \(i\)-th subsystem, \(y_{i}=h_{i}(x_{i})\in\mathbb{R}^{p}\) is the subsystem's output. For example, for a formation of land vehicles, \(h_{i}\) corresponds to the coordinates of each vehicle. It is considered that \(p\) is the same for all subsystems. _Formation_ is defined in the coordinate system moving along with the _desired trajectory_\(y_{d}(s)\in\mathbb{R}^{p}\), where \(s\) is a certain parameter, and the orthonormal vector \(\mathcal{F}(s)=\big{[}\mathbf{e}_{1}(s),\ldots,\mathbf{e}_{p}(s)\big{]}\in \mathbb{R}^{p}\), specifying a moving RMS, start which lies at the point \(y_{d}(s)\). The formation \(F\) is thus specified by \(k\) by the points \(P_{1},\ldots,P_{k}\in\mathcal{F}(s)\), \(F=\{P_{1},\ldots,P_{k}\}\), where \(P_{i}=\sum_{j=1}^{p}\alpha_{ij}\mathbf{e}_{j}\), and in the general case it changes over time. The key parameter is defined as the action reference \(s\), on the basis of which the desired trajectory \(y_{d}(s)\) is formed. To calculate the reference action, the reference projections are used, which transform the sensor data into a reference action \(s\), on the basis of which the time is then calculated, which is used to coordinate lower-level feedbacks. The synthesis of the control law consists of several steps. On the first of them, given \(y_{d}(s)\) and \(\{P_{1},\ldots,P_{k}\}\), the desired trajectory \(y_{d\,i}(s)=y_{d}(s)+\sum_{j=1}^{p}\alpha_{ij}\mathbf{e}_{j}(s)\) for each agent (subsystem) in the formation is found. The speed of the formation along the trajectory \(y_{d}(s)\) is specified by means of a strictly increasing function \(s=v(t)\). The formation control law \(u=u(x)\) must ensure the fulfillment of the control goal as \(\lim\limits_{t\to\infty}\big{(}y_{i}(t)-y_{d\,i}\big{(}v(t)\big{)}\big{)}=0\). With this control, there is an initial state \(x_{0}=[x_{01},\ldots,x_{0k}]^{T}\) such that the corresponding trajectory \(x_{di}(s)\) coincides with the given one, that is, \(h_{i}(x_{di}(s))=y_{di}(t)\). In the second step, the control law \(u_{i}=u_{i}(x,t)\) is synthesized separately for each subsystem using existing tracking methods ( signal tracking) or movement along a given trajectory ( path following). Different methods of controller synthesis can be used for different subsystems.