Skip to main content
SpringerLink
Log in

Necessary and Sufficient Conditions for Internal Stability of Linear Formations

  • CONTROL PROCESSES
  • Published:
Doklady Mathematics Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. T. Balch and R. C. Arkin, IEEE Trans. Rob. Autom. 14 (6), 926–939 (1998).

    Article  Google Scholar 

  2. H. G. Tanner and G. J. Pappas, “Formation input-to-state stability,” Proceedings of the 15th IFAC World Congress on Automatic Control (Barcelona, 2002), pp. 1512–1517.

  3. H. G. Tanner, V. Kumar, and G. J. Pappas, “Stability properties of interconnected vehicles,” Proceedings of the 15th International Symposium on Mathematical Theory of Networks and Systems, South Bend, Indiana (2002), paper no. 4615-2, pp. 1–12.

  4. H. G. Tanner, G. J. Pappas, and V. Kumar, “Input-to-state stability on formation graphs,” Proceedings of the 41th IEEE Conference on Decision and Control, Las Vegas, NV (2002), pp. 2439–2444.

  5. H. G. Tanner, G. J. Pappas, and V. Kumar, IEEE Trans. Rob. Autom. 20 (3), 443–455 (2004).

    Article  Google Scholar 

  6. K. K. Oh, M. C. Park, and H. S. Ahn, Automatica 53, 424–440 (2015).

    Article  Google Scholar 

  7. J. Lu, F. Chen, and G. Chen, Automatica 64, 112–120 (2016).

    Article  Google Scholar 

  8. E. D. Sontag, IEEE Trans. Autom. Control. 34 (4), 435–443 (1989).

    Article  Google Scholar 

  9. E. D. Sontag and Y. Wang, Syst. Control Lett. 24 (5), 351–359 (1995).

    Article  Google Scholar 

  10. S. N. Dashkovskiy, D. V. Efimov, and E. D. Sontag, Autom. Remote Control 72 (8), 1579–1614 (2011).

    Article  MathSciNet  Google Scholar 

  11. S. N. Vassilyev, R. I. Kozlov, and S. A. Ul’yanov, Proc. Steklov Inst. Math. 268, suppl. 1, 264–282 (2010).

    Article  MathSciNet  Google Scholar 

  12. S. N. Vassilyev, R. I. Kozlov, and S. A. Ul’yanov, Dokl. Math. 89 (2), 257–262 (2014).

    Article  MathSciNet  Google Scholar 

  13. S. Ul’yanov and N. Maksimkin, Math. Eng. Sci. Aerosp. 10 (3), 487–500 (2019).

    Google Scholar 

  14. H. K. Khalil, Nonlinear Systems (Prentice Hall, New Jersey, 2002).

    MATH  Google Scholar 

  15. F. Harary, Graph Theory (Addison-Wesley, Reading, Mass., 1969).

    Book  Google Scholar 

Download references

Funding

This work was supported in part by the Russian Foundation for Basic Research, grant nos. 19-08-00746 and 19-01-00301.

Author information

Authors and Affiliations

  1. Matrosov Institute for System Dynamics and Control Theory, Siberian Branch, Russian Academy of Sciences, 664033, Irkutsk, Russia

    A. V. Lakeyev

Authors
  1. A. V. Lakeyev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. V. Lakeyev.

Additional information

Translated by I. Ruzanova

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lakeyev, A.V. Necessary and Sufficient Conditions for Internal Stability of Linear Formations. Dokl. Math. 101, 71–75 (2020). https://doi.org/10.1134/S1064562420010172

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1064562420010172

Keywords:

Search

Navigation