# Modeling and Control of Satellite Formations: A Survey
## 4 Leader-Follower Formation Architecture
The theory assumes small relative distances between satellites, which is equally true for all elliptical orbits, as well as for the special case of a circular orbit due to the use of nonsingular orbital elements. An analytical filter has been introduced, which compensates for short-period changes in the relative state without using built-in digital filters, which are usually used to filter noise in the design of control systems. The application of this filter to maintain the formation on a given relative trajectory is considered. In [122], orbital elements are defined as a set of parameters that characterize the relative motion of a satellite formation based on the principle of geostationary spacecraft control. From these parameters, the orbital motion and maneuvering equations are derived. In addition, in [122], acceleration control laws are proposed, including out-of-plane and in-plane control. To demonstrate them as orbital maneuvers, the following are considered: the creation of a formation of two satellites, its reconfiguration, maneuvering at long distances, and the preservation of the formation. The results of modeling the dynamics of the movement of the formation under the influence of the gravitational field and eccentricity when applying the proposed law of control with feedback are presented, which show the effectiveness of the proposed method and the possibility of correct control of the slave satellite from the lead satellite. In [123], simple and general analytical solutions are considered for the optimal reconfiguration of a satellite constellation controlled by various linear dynamic equations. The calculus of variations is used for the analytical search for optimal trajectories and control. The proposed method makes it possible to predict in advance the exact form of optimal solutions without the need to solve the problem. It is shown that the optimal thrust vector as a function of the fundamental matrix of the given equations of state. The analytical solutions presented in this paper can be applied to most problems of satellite reconfiguration governed by linear dynamic equations. Numerical modeling confirms the brevity and accuracy of the solutions obtained. During the LISA missions [9], it may be necessary to reconfigure the formation to maintain or improve its functionality [10]. This reconfiguration can be described by relative dynamics, as presented, for example, in [23, 124]. Using relative dynamics in projection onto a circular orbit, the authors of [17] found analytical solutions for optimal two-pulse trajectories of reconfiguration of a formation of two spacecraft in orbit. The desired formation is characterized by nonsingular different orbital elements. To achieve the desired orbital-element differences, [17] proposes an analytical two-pulse solution found using the Gaussian variational equations. The solutions obtained can be easily implemented using onboard computing devices. In order to maximize mission life during reconfiguration, many papers consider the problem of optimization in terms of fuel consumption. The works by [PERSON] and his colleagues significantly developed a solution to the problem of control synthesis for impulsive systems, based on generalizations of the Hamilton-Jacobi-Bellman variational inequalities, which made it possible, within the framework of a single formalization, to study control problems for hybrid systems containing jump-like rearrangements of states [125, 126]. Optimization methods based on the calculus of variations, including the theory of basis vectors, encounter difficulties in finding the global optimal solution, since these methods can only search in a convex neighborhood around the initial approximation. Therefore, the global optimality of the solution is not guaranteed, especially with substantially non-convex objective functions. To mitigate these difficulties, several methods have been proposed, one of which is the \"hybrid optimization\"--a combination of the calculus of variations and global extremum search [127, 128].