Nonlinear Dynamics

, Volume 84, Issue 2, pp 977–990 | Cite as

Effect compensation of the presence of a time-dependent interior delay on the stabilization of the rotating disk-beam system

  • Boumediene Chentouf
Original Paper


In this article, we investigate the compensation problem of the effect of the interior time-dependent delay on the stabilization of a rotating disk-beam system. The physical system consists of a flexible beam free at one end, and attached to the center of the rotating disk whose angular velocity is time-varying. Assuming that a time-dependent interior delay is present in the system, we introduce a dynamic boundary force control at the free end of the beam and a torque control on the disk. Then, we show the destabilizing effect of the delay is compensated. Specifically, it is shown that the presence of such proposed controls assures the exponential stability of the system, provided that some reasonable conditions on the angular velocity of the disk and delay are fulfilled. Numerical examples in the case of constant delay are also provided to highlight the stability result.


Rotating disk-beam Interior control Time-dependent delay Dissipative boundary force control Torque control Stability 

List of symbols

\(\ell \)

Length of the beam

\(\rho \)

Mass per unit length of the beam


Flexural rigidity of the beam

\(I_\mathrm{d} \)

Disk’s moment of inertia


Beam’s displacement at time t with respect to the spatial variable x

\(\omega (t)\)

Angular velocity of the disk at time t

\(\mathcal{U}_\mathrm{I} (t)\)

Interior control

\(\mathcal{U}_\mathrm{F} (t)\)

Boundary force control


Torque control

\(\alpha \ge 0\)

Feedback gain of the interior control

\(\beta >0\)

Force control feedback gain

\(\gamma >0\)

Torque control feedback gain



The author would like to thank the Associate Editor and the anonymous referees for their constructive comments and valuable suggestions which have led to an improved version of this paper. Also, the author is grateful to one of the referees for having mentioned the recent articles [14, 15, 16, 17, 18]. Finally, the author acknowledges the support of Sultan Qaboos University.


  1. 1.
    Baillieul, J., Levi, M.: Rotational elastic dynamics. Physica 27, 43–62 (1987)MathSciNetMATHGoogle Scholar
  2. 2.
    Bloch, A.M., Titi, E.S.: On the dynamics of rotating elastic beams. In: Conte, Perdon, Wyman (eds.) Proceedings of the Conference on New Trends System Theory, Genoa, Italy. Birkhäuser, Cambridge, MA (1990)Google Scholar
  3. 3.
    Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitex, Springer, Berlin (2011)MATHGoogle Scholar
  4. 4.
    Brogliato, B., Lozano, R., Maschke, B., Egeland, O.: Dissipative Systems Analysis and Control: Theory and Applications. Springer, London (2007)CrossRefMATHGoogle Scholar
  5. 5.
    Chentouf, B., Couchouron, J.F.: Nonlinear feedback stabilization of a rotating body-beam without damping. In: ESAIM: COCV, vol. 4, pp. 515–535 (1999)Google Scholar
  6. 6.
    Chentouf, B., Wang, J.M.: Stabilization and optimal decay rate for a non-homogeneous rotating body-beam with dynamic boundary controls. J. Math. Anal. Appl. 318, 667–691 (2006)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Chentouf, B.: Stabilization of the rotating disk-beam system with a delay term in boundary feedback. Nonlinear Dyn. 78(3), 2249–2259 (2014)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chentouf, B.: Stabilization of a nonlinear rotating flexible structure under the presence of time-dependent delay term in a dynamic boundary control. IMA J. Math. Control Inf. doi: 10.1093/imamci/dnu044
  9. 9.
    Chentouf, B.: Compensation of the interior delay effect for a rotating disk-beam system. IMA J. Math. Control Inf. doi: 10.1093/imamci/dnv018
  10. 10.
    Chicone, C., Latushkin, Y.: Evolution Semigroups in Dynamical Systems and Differential Equations. Mathematical Surveys and Monographs, vol. 70. American Mathematical Society, Providence, RI (1999)Google Scholar
  11. 11.
    Coron, J.-M., d’Andréa-Novel, B.: Stabilization of a rotating body-beam without damping. IEEE Trans. Autom. Control 43(5), 608–618 (1998)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Dafermos, C.M.: Asymptotic stability in viscoelasticity. Arch. Ration. Mech. Anal. 37, 297–308 (1970)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Fridman, E.E., Nicaise, S., Valein, J.: Stabilization of second order evolution equations with unbounded feedback with time-dependent delay. SIAM J. Control Optim. 48(8), 5028–5052 (2010)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    He, W., Ge, S.S., How, B.V.E., Choo, Y.S., Hong, K.S.: Robust adaptive boundary control of a flexible marine riser with vessel dynamics. Automatica 47, 722–732 (2011)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    He, W., Ge, S.S.: Vibration control of a nonuniform wind turbine tower via disturbance observer. IEEE/ASME Trans. Mechatron. 20, 237–244 (2015)CrossRefGoogle Scholar
  16. 16.
    He, W., Sun, C., Ge, S.S.: Top tension control of a flexible marine riser by using integral-barrier Lyapunov function. IEEE/ASME Trans. Mechatron. 20, 497–505 (2015)CrossRefGoogle Scholar
  17. 17.
    He, W., Ge, S.S.: Vibration control of a flexible beam with output constraint. IEEE Trans. Ind. Electron. 2, 5023–5030 (2015)CrossRefGoogle Scholar
  18. 18.
    Jin, F.F., Guo, B.Z.: Lyapunov approach to output feedback stabilization for the Euler Bernoulli beam equation with boundary input disturbance. Automatica 52, 95–102 (2015)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Kato, T.: Linear evolution equations of “hyperbolic” type. J. Fac. Sci. Univ. Tokyo Sect I. 17, 241–258 (1970)MathSciNetMATHGoogle Scholar
  20. 20.
    Kato, T.: Linear evolution equations of “hyperbolic” type. II. J. Math. Soc. Jpn. 25, 648–666 (1973)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Kato, T.: Linear and quasilinear equations of evolution of hyperbolic type. In: da Prato, G., Geymonat, G. (eds.) Hyperbolicity: Lectures Given at a Summer School of the Centro Internazionale Matematico Estivo (C.I.M.E.), Cortona (Arezzo), Italy, Volume 2, pp. 125–191. Liguori Editore (1977)Google Scholar
  22. 22.
    Kato, T.: Abstract Differential Equations and Nonlinear Mixed Problems, Lezioni Fermiane [Fermi Lectures]. Scuola Normale Superiore, Pisa (1985)Google Scholar
  23. 23.
    Khalil, H.K.: Nonlinear Systems, 3rd edn. Prentice-Hall, Englewood Cliffs (2002)MATHGoogle Scholar
  24. 24.
    Laousy, H., Xu, C.Z., Sallet, G.: Boundary feedback stabilization of a rotating body-beam system. IEEE Trans. Autom. Control 41, 241–245 (1996)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Morgül, O.: Orientation and stabilization of a flexible beam attached to a rigid body: planar motion. IEEE Trans. Autom. Control 36, 953–963 (1991)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Morgül, O.: Constant angular velocity control of a rotating flexible structure. In: Proceedings of the 2nd Conference, ECC’93, pp. 299–302, Groningen, Netherlands (1993)Google Scholar
  27. 27.
    Morgül, O.: An exponential stability result for the wave equation. Automatica 38, 731–735 (2002)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Nicaise, S., Pignotti, C., Valein, J.: Stability of the heat and of the wave equations with boundary time-varying delays. Discret. Contin. Dyn. Syst. Ser. S 4, 693–722 (2011)MathSciNetMATHGoogle Scholar
  29. 29.
    Nicaise, S., Valein, J., Fridman, E.: Stability of the heat and of the wave equations with boundary time-varying delays. Discret. Contin. Dyn. Syst. Ser. S 2, 559–581 (2009)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)CrossRefMATHGoogle Scholar
  31. 31.
    Shang, Y.F., Xu, G.Q., Chen, Y.L.: Stability analysis of Euler–Bernoulli beam with input delay in the boundary control. Asian J. Control 14, 186–196 (2012)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Xu, C.Z., Sallet, G.: Boundary stabilization of a rotating flexible system. In: Curtain, R.F., Bensoussan, A., Lions, J.L. (eds.) Lecture Notes in Control and Information Sciences, vol. 185, pp. 347–365. Springer, New York (1992)Google Scholar
  33. 33.
    Xu, C.Z., Baillieul, J.: Stabilizability and stabilization of a rotating body-beam system with torque control. IEEE Trans. Autom. Control 38, 1754–1765 (1993)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsSultan Qaboos UniversityAl-Khod, MuscatOman

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