Geodesic
Necessary and sufficient conditions for internal stability of linear formations
Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 490 (2020), pp. 85-90.

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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.
Keywords: formation, input-to-state stability, acyclic digraph, Hurwitz matrix, stabilizability. 
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A.V.Lakeev. Necessary and sufficient conditions for internal stability of linear formations. Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 490 (2020), pp. 85-90. http://geodesic.mathdoc.fr/item/DANMA_2020_490_a19/

[1] Balch T., Arkin R.C., “Behavior-Based Formation Control for Multirobot”, IEEE Transactions on Robotics and Automation, 14:6 (1998), 926–939 | DOI

[2] Tanner H.G., Pappas G.J., “Formation input-to-state stability”, Proc. 15th IFAC World Congr. Autom. Control (Barcelona, 2002), 1512–1517

[3] Tanner H.G., Kumar V., Pappas G.J., “Stability properties of interconnected vehicles”, Proc. 15th International Symposium on Mathematical Theory of Networks and Systems (South Bend, Indiana, 2002), 4615-2, 1–12 (CD-ROM) | MR

[4] Tanner H.G., Pappas G.J., Kumar V., “Input-to-state Stability on Formation Graphs”, Proc. 41st IEEE Conference on Decision and Control (Las Vegas, NV, 2002), 2439–2444

[5] Tanner H.G., Pappas G.J., Kumar V., “Leader-to-formation stability”, IEEE Transactions on Robotics and Automation, 20:3 (2004), 443–455 | DOI

[6] Oh K.K., Park M.C., Ahn H.S., “A survey of multi-agent formation control”, Automatica, 53 (2015), 424–440 | DOI | MR | Zbl

[7] Lü J., Chen F., Chen G., “Nonsmooth leader-following formation control of nonidentical multi-agent systems with directed communication topologies”, Automatica, 64 (2016), 112–120 | DOI | MR | Zbl

[8] Sontag E.D., “Smooth stabilization implies coprime factorization”, IEEE Trans. Automat. Control, 34:4 (1989), 435–443 | DOI | MR | Zbl

[9] Sontag E.D., Wang Y., “On characterizations of the input-to-state stability property”, Systems Control Letters, 24:5 (1995), 351–359 | DOI | MR | Zbl

[10] Dashkovskii S.N., Efimov D.V., Contag E.D., “Ustoichivost ot vkhoda k sostoyaniyu i smezhnye svoistva sistem”, AiT, 2011, no. 8, 3–40 | MR

[11] Vasilev S.N., Kozlov R.I., Ulyanov S.A., “Analiz koordinatnykh i drugikh preobrazovanii modelei dinamicheskikh sistem metodom reduktsii”, Trudy instituta matematiki i mekhaniki UrO RAN, 15, no. 3, 2009, 38–55

[12] Vasilev S.N., Kozlov R.I., Ulyanov S.A., “Ustoichivost mnogorezhimnykh formatsii”, DAN, 455:3 (2014), 269–274

[13] Ul'yanov S., Maksimkin N., “Formation path-following control of multi-AUV systems with adaptation of reference speed”, Mathematics in Engineering, Science and Aerospace, 10:3 (2019), 487–500 | MR

[14] Khalil X.K., Nelineinye sistemy, RKhD, M., 2009

[15] Kharari F., Teoriya grafov, Mir, M., 1973