Necessary and sufficient conditions for internal stability of linear formations

Necessary and sufficient conditions for the internal stability of formations whose dynamics are defined by linear differential equations have been obtained. The classes of admissible controls are specified as programmed controls for leaders and affine feedback controls depending on the object and leader states for followers. The conditions obtained are easy to verify and consist of (i) the stabilizability of a pair of matrices for the follower equations, (ii) the Hurwitz property and (iii) the coincidence of matrices for leaders in the multi-leader case, and (iv) the solvability of some linear equations and equality constraints on the vectors defining the desired relative leader–follower positions. Furthermore, the entire class of controls ensuring linear internal stability is described. By using the conditions obtained, it is shown that, in fact, only single-leader formations can possess internal stability. In the class of single-leader formations, a subclass of formations (whose graph is an input tree) is identified that are free of equality constraints, which are the main obstacle to the internal stability of multi-leader formations.