Optimization Letters

, Volume 13, Issue 8, pp 1937–1952 | Cite as

LMI and SDP technique for stability analysis of nonlinear delay systems subject to constraints

  • N. SedovaEmail author
  • I. Pertseva
Original Paper


The aim of the study is to present an estimation of the domain of attraction (DA) for nonlinear delay systems. The proposed approach is based on the assumption that on a subset of instant states the system is represented by a continuous-time Takagi–Sugeno (TS) system with delay. Because of the constraints defining the subset under consideration, an important issue is to estimate the set of initial points such that the trajectories starting at those points remain to satisfy the constraints. Using Lyapunov functions with the Razumikhin technique, we describe the problem of constructing an inner estimation of the DA in terms of invariant ellipsoids. As the result, the problem reduces to linear matrix inequalities and semidefinite programs which can be solved numerically. The method is also applied to constructing some outer estimate for the attractor of a nonlinear delay system subject to asymptotic stability for an approximating TS system. The resulting ellipsoidal estimate for the attractor depends on the estimate of the approximation error.


Nonlinear delay system Stability Takagi–Sugeno system System with constraints LMI SDP Razumikhin technique 


  1. 1.
    La Salle, J., Lefschetz, S.: Stability by Liapunov’s Direct Method with Applications. Academic Press, New York (1961)zbMATHGoogle Scholar
  2. 2.
    Kamenetskii, V.A.: Parametric stabilization of nonlinear control systems under state constraints. Autom. Remote Control 57(10), 1427–1435 (1996)MathSciNetGoogle Scholar
  3. 3.
    Polyak, B., Shcherbakov, P.: Ellipsoidal approximations to attraction domains of linear systems with bounded control. In: American Control Conference, pp. 5363–5367 (2009)Google Scholar
  4. 4.
    Prasolov, A.V.: On an estimate of the domain of attraction for systems with after effect. Ukr. Math. J. 50(5), 761–769 (1998)MathSciNetCrossRefGoogle Scholar
  5. 5.
    González, T., Bernal, M.: Progressively better estimates of the domain of attraction for nonlinear systems via piecewise Takagi–Sugeno models: stability and stabilization issues. Fuzzy Sets Syst. 297, 73–95 (2016)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ting, C.-S.: A robust fuzzy control approach to stabilization of nonlinear time-delay systems with saturating inputs. Int. J. Fuzzy Syst. 10(1), 50–60 (2008)MathSciNetGoogle Scholar
  7. 7.
    Yang, R.-M., Wang, Y.-Z.: Stability analysis and estimate of domain of attraction for a class of nonlinear systems with time-varying delay. Acta Autom. Sin. 38(5), 716–723 (2012)CrossRefGoogle Scholar
  8. 8.
    Andreev, A.S.: The Lyapunov functionals method in stability problems for functional differential equations. Autom. Remote Control 70(9), 1438–1486 (2009)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hale, J.: Theory of Functional Differential Equations. Springer, Berlin (1977)CrossRefGoogle Scholar
  10. 10.
    Leng, S., Lin W., Kurths, J.: Basin stability in delayed dynamics. Nature Scientific Reports, vol. 6 (2016).
  11. 11.
    Boyd, S., El Ghaoui, L., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory, vol. 15. SIAM, Philadelphia (1994)CrossRefGoogle Scholar
  12. 12.
    Nesterov, Y., Nemirovski, A.: Interior Point Polynomial Methods in Convex Programming: Theory and Applications. SIAM, Philadelphia (1994)CrossRefGoogle Scholar
  13. 13.
    Tanaka, K., Wang, H.O.: Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach. Wiley, New York (2001)CrossRefGoogle Scholar
  14. 14.
    Khlebnikov, M.V., Polyak, B.T., Kuntsevich, V.M.: Optimization of linear systems subject to bounded exogenous disturbances: the invariant ellipsoid technique. Autom. Remote Control 72(11), 2227–2275 (2011)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Fridman, E., Shaked, U.: An ellipsoid bounding of reachable systems with delay and bounded peak inputs. IFAC Proc. Vol. 36(19), 269–274 (2003)CrossRefGoogle Scholar
  16. 16.
    Sontag, E. D.: Stability and stabilization: discontinuities and the effect of disturbances. In: Nonlinear analysis, differential equations and control (Eds. Clarke, F.H. and Stern, R.J.) Springer, Dordrecht, pp. 551–598 (1999)CrossRefGoogle Scholar
  17. 17.
    Sedova, N.O.: On the principle of reduction for the nonlinear delay systems. Autom. Remote Control 72(9), 1864–1875 (2011)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Fantuzzi, C., Rovatti, R.: On the approximation capabilities of the homogeneous Takagi–Sugeno model, fuzzy systems, 1996. In: Proceedings of the Fifth IEEE International Conference on, vol. 2 (1996)Google Scholar
  19. 19.
    Buckley, J.: Universal fuzzy controllers. Automatica (J. IFAC) 28(6), 1245–1248 (1992)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Wang, L.-X., Mendel, J.M.: Fuzzy basis functions, universal approximation, and orthogonal least-squares learning. IEEE Trans. Neural Netw. 3(5), 807–814 (1992)CrossRefGoogle Scholar
  21. 21.
    Egrashkina, J.E., Sedova, N.O.: On approximate Takagi–Sugeno linear representations of nonlinear functions. Matem. Mod. 29(1), 20–32 (2017). [in Russian]zbMATHGoogle Scholar
  22. 22.
    Razumikhin, B.S.: On stability on systems with delay. Prikl. Mat. Mekh. 20, 500–512 (1956) (in Russian) Google Scholar
  23. 23.
    Druzhinina, O.V., Sedova, N.O.: Analysis of stability and stabilization of cascade systems with time delay in terms of linear matrix inequalities. J. Comput. Syst. Sci. Int. 56(1), 19–32 (2017)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Sedova, N.O.: The design of digital stabilizing regulators for continuous systems based on the Lyapunov function approach. Autom. Remote Control 73(10), 1734–1743 (2012)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Xie, X.-P., Liu, Z.-W., Zhu, X.-L.: An efficient approach for reducing the conservatism of LMI-based stability conditions for continuous-time TS fuzzy systems. Fuzzy Sets Syst. 263, 71–81 (2015)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Information, and Aviation TechnologiesUlyanovsk State UniversityUlyanovskRussia

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