Some Recent Results on Direct Delay-Dependent Stability Analysis: Review and Open Problems

  • Libor PekařEmail author
  • Pavel Navrátil
  • Radek Matušů
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 765)


This contribution focuses an overview of selected results on time-delay systems stability analysis in the delay space, recently published in outstanding high-impacted journals and top conferences and meetings. A numerical gridding algorithm solving this problem designed by the first author is included as well. The theoretical background and a concise literature overview are followed by the list of practical and software applications. Unsolved tasks and open problems stemming from the analysis of presented methods and results concisely conclude the paper. The reader is supposed to use this survey to follow some of the presented techniques in his/her own research or engineering practice.


Delay-dependent stability Engineering application Survey Time-delay systems 



This work was performed with the financial support by the Minis-try of Education, Youth and Sports of the Czech Republic within the National Sustain-ability Programme project No. LO1303 (MSMT-7778/2014).


  1. 1.
    Li, X., Gao, H., Gu, K.: Delay-independent stability analysis of linear time-delay systems based on frequency discretization. Automatica 70(3), 288–294 (2016)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Hertz, D., Jury, E.I., Zeheb, E.: Stability independent and dependent of delay for delay differential systems. J. Franklin Inst. 318(3), 143–150 (1984)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Xu, S., Lam, J.: Improved delay-dependent stability criteria for time delay systems. IEEE Trans. Autom. Control 50(3), 384–387 (2005)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Sipahi, R., Olgac, N.: Complete stability robustness of third-order LTI multiple time-delay systems. Automatica 41(8), 1413–1422 (2005)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Pepe, P., Jiang, Z.P.: A Lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems. Syst. Control Lett. 55(12), 1006–1014 (2006)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Pekař, L., Matušů, R., Prokop, R.: Gridding discretization-based multiple stability switching delay search algorithm: The movement of a human being on a controlled swaying bow. PLoS ONE 12(6), e0178950 (2017)CrossRefGoogle Scholar
  7. 7.
    Pekař, L., Prokop, R.: Direct stability-switching delays determination procedure with differential averaging. Trans. Inst. Meas. Control (2017).
  8. 8.
    Gu, K., Kharitonov, V.L., Chen, J.: Stability of Time-Delay Systems. Birkhäuser, Basel (2003)CrossRefGoogle Scholar
  9. 9.
    Sipahi, R., Delice, I.I.: On some features of core hypersurfaces related to stability switching of LTI systems with multiple delays. IMA J. Math. Control Inf. 31(2), 257–272 (2014)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Castanos, F., Estrada, E., Mondié, S., Ramírez, A.: Passivity-based PI control of first-order systems with I/O communication delays: a frequency domain analysis. Int. J. Control. in press
  11. 11.
    Sipahi, R.: Design of imaginary spectrum of LTI systems with delays to manipulate stability regions. In: Insperger, T., Ersal, T., Orosz, G. (eds.) Time-Delay Systems: Theory, Numerics, Applications, and Experiments, pp. 127–140. Springer, New York (2017)CrossRefGoogle Scholar
  12. 12.
    Kammer, A.S., Olgac, N.: Non-conservative stability assessment of LTI dynamics with distributed delay using CTCR paradigm. In: 2015 American Control Conference, pp. 4597–4602. Palmer House Hilton, Chicago (2015)Google Scholar
  13. 13.
    Gao, Q., Zalluhoglu, U., Olgac, N.: Investigation of local stability transitions in the spectral delay space and delay space. J. Dyn. Syst. Meas. Control 136(5) (2014). ASME, Article no. 051011CrossRefGoogle Scholar
  14. 14.
    Gao, Q., Olgac, N.: Stability analysis for LTI systems with multiple time delays using the bounds of its imaginary spectra. Syst. Control Lett. 102, 112–118 (2017)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Fazelinia, H., Sipahi, R., Olgac, N.: Stability robustness analysis of multiple time-delayed systems using ‘Building Block’ concept. IEEE Trans. Autom. Control 52(5), 799–810 (2007)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Cepeda-Gomez, R.: Finding the exact delay bound for consensus of linear multi-agent systems. Int. J. Syst. Sci. 47(11), 2598–2606 (2016)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Walton, K.E., Marshall, J.E.: Direct method for TDS stability analysis. IEE Proc. D-Control Theory Appl. 134(2), 101–107 (1987)CrossRefGoogle Scholar
  18. 18.
    Sönmez, Ş., Ayasun, S., Nwankpa, C.O.: An exact method for computing delay margin for stability of load frequency control systems with constant communication delays. IEEE Trans. Power Syst. 31(1), 370–377 (2016)CrossRefGoogle Scholar
  19. 19.
    Li, X.-G., Niculescu, S.-I., Ҫela, A., Wang, H.-H., Cai, T.-Y.: On τ-decomposition frequency-sweeping test for a class of time-delay systems, Part II: Multiple roots case. IFAC Proc. Vol. 45(14), 138–143 (2012)CrossRefGoogle Scholar
  20. 20.
    Li, X.-G., Niculescu, S.-I., Ҫela, A., Wang, H.-H., Cai, T.-Y.: On computing Puiseux series for multiple imaginary characteristic roots of LTI systems with commensurate delays. IEEE Trans. Autom. Control 58(5), 1338–1343 (2013)CrossRefGoogle Scholar
  21. 21.
    Ma, J., Zheng, B., Zhang, C.: A matrix method for determining eigenvalues and stability of singular neutral delay-differential systems. J. Appl. Math. 2012 (2012). Article ID 749847Google Scholar
  22. 22.
    Ochoa, G., Kharitonov, V.L., Modié, S.: Critical frequencies and parameters for linear delay systems: a Lyapunov matrix approach. Syst. Control Lett. 63(9), 781–790 (2013)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Cao, J.: Improved delay-dependent exponential stability criteria for time-delay system. J. Franklin Inst. 350(4), 790–801 (2013)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Xu, Q., Wang, Z.: Exact stability test of neutral delay differential equations via a rough estimation of the testing integral. Int. J. Dyn. Control 2(2), 154–163 (2014)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Xu, Q., Stépán, G., Wang, Z.: Delay-dependent stability analysis by using delay-independent integral evaluation. Automatica 70(3), 153–157 (2016)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Domoshnitsky, A., Maghakyan, A., Berezansky, L.: W-transform for exponential stability of second order delay differential equations without damping terms. J. Inequal. Appl. 2017(1) (2017). Article no. 20Google Scholar
  27. 27.
    Pekař, L.: Enhanced TDS stability analysis method via characteristic quasipolynomial polynomization. In: Šilhavý, R. et al. (eds.) Cybernetics and Mathematics Applications in Intelligent Systems: Proceedings of the 6th Computer Science On-line Conference 2017 (CSOC 2017), vol. 2, pp. 20–29. Springer, Heidelberg (2017)Google Scholar
  28. 28.
    Perng, J.-W.: Stability analysis of parametric time-delay systems based on parameter plane method. Int. J. Innov. Comput. Inf. Control 8(7A), 4535–4546 (2012)Google Scholar
  29. 29.
    Ramachandran, P., Ram, Y.M.: Stability boundaries of mechanical controlled system with time delay. Mech. Syst. Signal Process. 27, 523–533 (2012)CrossRefGoogle Scholar
  30. 30.
    Chen, J., Gu, G., Nett, C.N.: A new method for computing delay margins for stability of linear delay systems. Syst. Control Lett. 26(2), 107–117 (1995)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Mulero-Martínez, J.I.: Modified Schur-Cohn criterion for stability of delayed systems. Math. Probl. Eng. 2015 (2015). Article ID 846124MathSciNetGoogle Scholar
  32. 32.
    Gao, W., Ye, H., Liu, Y. Wang, L, Ci, W.: Comparison of three stability analysis methods for delayed cyber-physical power system. In: 2016 China International Conference on Electricity Distribution (CICED 2016), paper no. CP1252, Xi’an, China (2016)Google Scholar
  33. 33.
    Ye, H., Gao, W., Mou, Q., Liu, Y.: Iterative infinitesimal generator discretization-based method for eigen-analysis of large delayed cyber-physical power system. Electr. Pow. Syst. Res. 143, 389–399 (2017)CrossRefGoogle Scholar
  34. 34.
    Niu, X., Ye, H., Liu, Y., Liu, X.: Padé approximation based method for computation of eigenvalues for time delay power system. In: The 48th International Universities’ Power Engineering Conference, Dublin, Ireland, pp. 1–4 (2013)Google Scholar
  35. 35.
    Wu, Z., Michiels, W.: Reliably computing all characteristic roots of delay differential equations in a given right half plane using a spectral method. J. Comput. Appl. Math. 236(9), 2499–2514 (2012)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Kishor, N., Haarla, L., Purwar, S.: Stability analysis and stabilization of delayed reduced-order model of large electric power system. Int. Trans. Electr. Energy Syst. 26, 1882–1897 (2016)CrossRefGoogle Scholar
  37. 37.
    Olgac, N., Zulluhoglu, U., Kammer, A.S.: On blade/casing rub problems in turbomachinery: an efficient delayed differential equation approach. J. Sound Vib. 333, 6662–6675 (2014)CrossRefGoogle Scholar
  38. 38.
    Ai, B., Sentis, L., Paine, N., Han, S., Mok, A., Fok, C.-L.: Stability and performance analysis of time-delayed actuator control systems. J. Dyn. Syst. Meas. Control 138(5) (2016). ASME, Article no. 051005CrossRefGoogle Scholar
  39. 39.
    Ergenc, A.F., Olgac, N., Fazelinia, H.: Extended Kronecker summation for cluster treatment of LTI systems with multiple delays. SIAM J. Control Optim. 46(1), 143–155 (2007)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Alikoç, B., Mutlu, I., Ergenc, A.F.: Stability analysis of train following model with multiple communication delays. In: The 1st IFAC Workshop on Advances in Control and Automation Theory for Transportation Applications, Istanbul, Turkey, pp. 13–18 (2013)CrossRefGoogle Scholar
  41. 41.
    Eris, O., Ergenc, A.F.: Delay scheduling for delayed resonator applications. IFAC-PapersOnline 49(10), 77–81 (2016)CrossRefGoogle Scholar
  42. 42.
    Sipahi, R., Atay, F.M., Niculescu, S.-I.: Stability analysis of a constant time-headway driving strategy with driver memory effects modeled by distributed delays. IFAC-PapersOnline 48(12), 276–281 (2015)Google Scholar
  43. 43.
    Breda, D., Maset, S., Vermiglio, R.: TRACE-DDE: a tool for robust analysis and characteristic equations for delay differential equations. In: Loiseau, J., et al. (eds.) Topics in Time Delay Systems: Analysis, Algorithm and Control, pp. 145–155. Springer, Berlin (2009)CrossRefGoogle Scholar
  44. 44.
    Qiao, W., Sipahi, R.: Delay-dependent coupling for a multi-agent LTI consensus system with inter-agent delays. Physica D 267, 112–122 (2014)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Gölgeli, M., Özbay, H.: A mathematical model for cholesterol biosynthesis under nicotine exposure. IFAC-PapersOnline 49(10), 258–262 (2016)CrossRefGoogle Scholar
  46. 46.
    Avanessoff, D., Fioravanti, A.R., Bonnet, C.: YALTA: a Matlab toolbox for the H-stability analysis of classical and fractional systems with commensurate delays. IFAC Proc. Vol. 46(2), 839–844 (2013)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of Applied InformaticsTomas Bata University in ZlínZlínCzech Republic

Personalised recommendations