# Modeling and Control of Satellite Formations: A Survey
## 2 Multiple-Input-Multiple-Output Formation Architecture
The results of [63] can be used to find a trade-off between time and fuel consumption for missions that have time constraints, such as rescue missions or evasive maneuvers in outer space. The studies were continued in [64,66], where impulsive and time-fixed solutions with minimum fuel consumption for the problem of rendezvous and interception in orbit with internal trajectory constraints were obtained. Transitions between coplanar circular orbits in a gravitational field along a circular trajectory representing the minimum or maximum radius of the admissible orbit are considered. Basis vector theory has been extended by introducing trajectory constraints. In [64,66], Equations (18) and (19) are used, supplemented by the objective function to be minimized \(J=\int_{0}^{T}\Gamma(t)\mathrm{d}t\) corresponding fuel consumption. For the \(n\)-pulse solution, the thrust force (per unit mass) is described by the expression \(\Gamma=\sum_{k=1}^{n}\Delta v_{k}\delta(t-t_{k})\), where \(0\leqslant t_{1}\leqslant\ldots\leqslant t_{n}\leqslant T\) and \(\Delta v_{k}\delta(t-t_{k})\) represent impulses at times \(t_{k}\), expressed in terms of the \(\delta\) denotes the Dirac function. Therefore, the objective function takes the form \(J=\sum_{k=1}^{n}\Delta v_{k}\), where \(\Delta v_{k}\) are the discontinuities of the amplitudes of the velocity vector. In [64,66], a restriction of the inner minimum or maximum radius of the inner trajectory is introduced in the form \(r(t)\geqslant r_{\min}\) or \(r(t)\leqslant r_{\max}\) and is implemented by forcibly limiting the radius of the periapsis or apocenter. Initial orbit and restrictions on the place of meeting, or interception, are defined in terms of the constraint function \(\Psi\big{(}r(0),v(0),r(T),v(T)\big{)}=0\). Otherwise, without introducing restrictions, the problem statement is the same as in [63]. In [64,66] the optimal number of pulses is determined, as well as their time and moments of appearance. The existence of bounding boundary arcs is investigated, as well as the optimality of the class of singular arc solutions. In [66], a bifurcation phenomenon is discovered that can occur in an optimal solution for a constrained encounter problem. It consists of the fact that for symmetric boundary conditions, symmetric optimal solutions are expected and they found for sufficiently short transit times. However, when the transition time increases to some critical value, some of the extreme solutions become asymmetric and ambiguous, and the only symmetric extreme solution splits into two asymmetric ones. For the problem of controlling a formation of two satellites (according to the master-slave scheme), [68] (see also the review [69]) considers various methods for applying continuous control in sliding modes, including second-order sliding mode algorithms [70], for example, super-twisting algorithm, continuous sliding mode control with a sliding mode disturbance observer (SMDO) [71], algorithms with integral sliding surface (ISS) [72]. Pulse Width Modulation (PWM) is used to convert continuous control signals into pulse trains that can be implemented using satellite jet engines. A numerical example is considered in which the position of the slave satellite relative to the master is described on the appropriate time scale by the following equations  
\[ \begin{cases}\ddot{x}-2\dot{y}-3x=u_{x}+d_{x},\\ \dot{y}+2\dot{y}-3x=u_{y}+d_{y},\\ \ddot{x}+z=u_{z}+d_{z},\end{cases}  \tag{27}\]  
where \(x,y,z\) are the coordinates of the slave satellite relative to the master \(u_{x}\), \(u_{y}\), \(u_{z}\) are the control forces acting on the slave satellite, \(d_{x}\), \(d_{y}\), \(d_{z}\) includes various perturbations acting on a two-satellite system, caused, for example, by the non-centrality of the gravitational field, solar pressure, lunar-solar tides, the rotational-vibrational motion of the Earth in space, oscillations of the Earth's pole, uneven rotation of the Earth, and also the deviation linearization error position.