Skip to main content

Scanning the Space of Parameters for Stability Regions of a Class of Time-Delay Systems; A Lyapunov Matrix Approach

  • 399 Accesses

Part of the Advances in Delays and Dynamics book series (ADVSDD,volume 10)

Abstract

A computationally efficient approach for carrying out the second step in the production of stability charts for a class of time-delay systems is presented. The methodology consists of detecting stable regions in the delay-parameter space based on a stability test in terms of the delay Lyapunov matrix, which plays a central role in the framework of complete type Lyapunov–Krasovskii functionals. Our approach is tested on a number of challenging academic examples found in the literature, and it is also employed on the analysis of the effects of non-intentional input delays in delay-based control schemes.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-030-11554-8_10
  • Chapter length: 15 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   109.00
Price excludes VAT (USA)
  • ISBN: 978-3-030-11554-8
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Hardcover Book
USD   149.99
Price excludes VAT (USA)
Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Notes

  1. 1.

    Local stability analysis of non-linear systems follows from linearization, meaning that the technique presented here may be used as a first approximation to the stability analysis of non-linear dynamics.

  2. 2.

    It is worthy of mention that the delay Lyapunov matrix has exact solution only in the commensurate delays case, in the rest of the chapter, we take advantage of this characteristic to optimize the scanning process such that efficiency is improved.

  3. 3.

    That is, the characteristic equation of (1), namely \(\det (sI-\sum _{j=0}^mA_je^{-h_js})=0\), has no symmetric eigenvalues with respect to the imaginary axis in the complex plane.

References

  1. Breda, D., Maset, S., Vermiglio, R.: Pseudospectral differencing methods for characteristic roots of delay differential equations. SIAM J. Sci. Comput. 27, 482–495 (2005)

    MathSciNet  CrossRef  Google Scholar 

  2. Brethé, D., Loiseau, J.J.: An effective algorithm for finite spectrum assignment of single-input systems with delays. Math. Comput. Simul. 45(3), 339–348 (1998)

    MathSciNet  CrossRef  Google Scholar 

  3. Cuvas, C.: Contribución al estudio de la matriz de Lyapunov. Ph.D. thesis, CINVESTAV-IPN, México (2015)

    Google Scholar 

  4. Cuvas, C., Mondié, S.: Necessary stability conditions for delay systems with multiple pointwise and distributed delays. IEEE Trans. Autom. Control 61(7), 1987–1994 (2016)

    MathSciNet  CrossRef  Google Scholar 

  5. Egorov, A.V.: A new necessary and sufficient stability condition for linear time-delay systems. IFAC Proc. Vol. 47(3), 11018–11023 (2014)

    CrossRef  Google Scholar 

  6. Egorov, A.V., Mondié, S.: Necessary stability conditions for linear delay systems. Automatica 50(12), 3204–3208 (2014)

    MathSciNet  CrossRef  Google Scholar 

  7. Egorov, A.V., Cuvas, C., Mondié, S.: Necessary and sufficient stability conditions for linear systems with pointwise and distributed delays. Automatica 80, 218–224 (2017)

    MathSciNet  CrossRef  Google Scholar 

  8. Gu, K., Niculescu, S.I., Chen, J.: On stability crossing curves for general systems with two delays. J. Math. Anal. Appl. 311(12), 231–253 (2005)

    MathSciNet  CrossRef  Google Scholar 

  9. Insperger, T., Stépán, G.: Semi-discretization for Time-Delay Systems: Stability and Engineering Applications, vol. 178. Springer Science & Business Media (2011)

    Google Scholar 

  10. Kharitonov, V.L.: Time-Delay Systems: Lyapunov Functionals and Matrices. Birkhäuser, Basel (2013)

    CrossRef  Google Scholar 

  11. Manitius, A.Z., Olbrot, A.W.: Finite spectrum assignment for systems with delays. IEEE Trans. Autom. Control 24(4), 541–552 (1979)

    MathSciNet  CrossRef  Google Scholar 

  12. Mondié, S., Cuvas, C., Ramírez, A., Egorov, A.: Necessary conditions for the stability of one delay systems: a Lyapunov matrix approach. In: Proceedings of the 10th IFAC Workshop Time Delay Systems, pp. 1474–6670 (2012)

    Google Scholar 

  13. Neimark, J.: D-subdivisions and spaces of quasi-polynomials. Prikl. Mat. i Meh. 13, 349–380 (1949)

    Google Scholar 

  14. Olgac, N., Sipahi, R.: An exact method for the stability analysis of time-delayed linear time-invariant (LTI) systems. IEEE Trans. Autom. Control 47(5), 793–797 (2002)

    MathSciNet  CrossRef  Google Scholar 

  15. Ramírez, A.: Design of maximum exponential decay rate for LTI-SISO systems via delay-based controllers. Ph.D. thesis, CINVESTAV-IPN, México (2015)

    Google Scholar 

  16. Ramírez, A., Garrido, R., Mondié, S.: Integral retarded velocity control of DC servomotors. In: Proceedings of the 11th IFAC Workshop Time Delay Systems, pp. 558–563 (2013)

    Google Scholar 

  17. Ramírez, A., Garrido, R., Mondié, S.: Velocity control of servo systems using an integral retarded algorithm. ISA Trans. 58, 357–366 (2015)

    CrossRef  Google Scholar 

  18. Ramírez, A., Mondié, S., Garrido, R.: Proportional integral retarded control of second order linear systems. In: Proceedings of the 52nd IEEE Conference on Decision Control, pp. 2239–2244 (2013)

    Google Scholar 

  19. Ramírez, A., Mondié, S., Garrido, R., Sipahi, R.: Design of proportional-integral-retarded (PIR) controllers for second-order LTI systems. IEEE Trans. Autom. Control 61(6), 1688–1693 (2016)

    MathSciNet  CrossRef  Google Scholar 

  20. Roose, D., Luzyanina, T., Engelborghs, K., Michiels, W.: Software for stability and bifurcation analysis of delay differential equations and applications to stabilization. In: Niculescu, S.I., Gu K. (eds.) Advances in Time-Delay Systems, pp. 167-181. Springer (2004)

    Google Scholar 

  21. Santos, O., Mondié, S., Kharitonov, V.L.: Linear quadratic suboptimal control for time delays systems. Int. J. Control 82(1), 147–154 (2009)

    MathSciNet  CrossRef  Google Scholar 

  22. Sipahi, R., Delice, I.I.: Advanced clustering with frequency sweeping methodology for the stability analysis of multiple time-delay systems. IEEE Trans. Autom. Control 56(2), 467–472 (2011)

    MathSciNet  CrossRef  Google Scholar 

  23. Sipahi, R., Olgac, N.: A unique methodology for the stability robustness of multiple time delay systems. Syst. Control Lett. 55(10), 819–825 (2006)

    MathSciNet  CrossRef  Google Scholar 

  24. Sipahi, R., Fazelinia, H., Olgac, N.: Stability analysis of LTI systems with three independent delays. A computationally efficient procedure. J. Dyn. Syst. Meas. Control 131(5), 051013 (2009)

    CrossRef  Google Scholar 

  25. Sipahi, R., Niculescu, S.I., Abdallah, C.T., Michiels, W., Gu, K.: Stability and stabilization of systems with time delay. IEEE Control Syst. 31(1), 38–65 (2011)

    MathSciNet  CrossRef  Google Scholar 

  26. Stépán, G.: Retarded Dynamical Systems: Stability and Characteristic Functions. Wiley, New York (1989)

    MATH  Google Scholar 

  27. Toker, O., Özbay, H.: Complexity issues in robust stability of linear delay-differential systems. Math. Control Signals Syst. 9(4), 386–400 (1996)

    MathSciNet  CrossRef  Google Scholar 

  28. Vyhlidal, T., Zitek, P.: Mapping based algorithm for large-scale computation of quasi-polynomial zeros. IEEE Trans. Autom. Control 54(1), 171–177 (2009)

    MathSciNet  CrossRef  Google Scholar 

Download references

Acknowledgements

The presented research has been supported by Conacyt, grant 180725

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Sabine Mondié .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Verify currency and authenticity via CrossMark

Cite this chapter

Cuvas, C., Ramírez, A., Juárez, L., Mondié, S. (2019). Scanning the Space of Parameters for Stability Regions of a Class of Time-Delay Systems; A Lyapunov Matrix Approach. In: Valmorbida, G., Seuret, A., Boussaada, I., Sipahi, R. (eds) Delays and Interconnections: Methodology, Algorithms and Applications. Advances in Delays and Dynamics, vol 10. Springer, Cham. https://doi.org/10.1007/978-3-030-11554-8_10

Download citation