Automation and Remote Control

, Volume 73, Issue 6, pp 937–948 | Cite as

Settling time in a linear dynamic system with bounded external disturbances

  • M. V. Khlebnikov
Linear Systems


The article deals with the problem of estimating the settling time for a linear dynamic system subjected to the action of nonrandom bounded external disturbances. The suggested approach is based on the method of invariant ellipsoids and the technique of linear matrix inequalities. Both the continuous and the discrete version of the problem are considered. As an example the problem of control of a two-mass system is investigated.


Remote Control Linear Matrix Inequality External Disturbance Settling Time Optimal Controller 
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  1. 1.
    Nazin, S.A., Polyak, B.T., and Topunov, M.V., Rejection of Bounded Exogenous Disturbances by the Method of Invariant Ellipsoids, Autom. Remote Control, 2007, vol. 68, no. 3, pp. 467–486.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Balandin, D.V. and Kogan, M.M., Sintez zakonov upravleniya na osnove lineinykh matrichnykh neravenstv (Synthesis of Control Laws on the Basis of Linear Matrix Inequalities), Moscow: Fizmatlit, 2007.Google Scholar
  3. 3.
    Boyd, S., El Ghaoui, L., Feron, E., and Balakrishnan, V., Linear Matrix Inequalities in System and Control Theory, Philadelphia: SIAM, 1994.zbMATHCrossRefGoogle Scholar
  4. 4.
    Khlebnikov, M.V., Polyak, B.T., and Kuntsevich, V.M., Optimization of Linear Systems Subject to Bounded Exogenous Disturbances: The Invariant Ellipsoid Technique, Autom. Remote Control, 2011, vol. 72, no. 11, pp. 2227–2275.CrossRefGoogle Scholar
  5. 5.
    Nesterov, Yu.E., Metody vypukloi optimizatsii (Convex Optimization Methods), Moscow: MTsNMO, 2009.Google Scholar
  6. 6.
    Boyd, S. and Vandenberge, L., Convex Optimization, Cambridge: Cambridge Univ. Press, 2004.zbMATHGoogle Scholar
  7. 7.
    Letov, A.M., Matematicheskaya teoriya protsessov upravleniya (Mathematical Theory of Control Processes) Moscow: Nauka, 1981.Google Scholar
  8. 8.
    Khlebnikov, M.V., Evaluation of the Settling Time of Linear Dynamic Systems with Bounded Disturbances, Proc. II Traditional All-Russia Youth Summer School “Control, Information, and Optimization,” Pereslavl-Zalesskii, June 20–27, 2010, Moscow: Inst. Probl. Upravlen., 2010, pp. 152–164.Google Scholar
  9. 9.
    Khlebnikov, M.V., A Nonfragile Controller for Suppressing Exogenous Disturbances, Autom. Remote Control, 2010, vol. 71, no. 4, pp. 640–653.MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Abedor, J., Nagpal, K., and Poolla, K., A Linear Matrix Inequality Approach to Peak-to-Peak Gain Minimization, Int. J. Rob. Nonlinear Control, 1996, vol. 6, pp. 899–927.MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Reinelt, W., Robust Control of a Two-Mass-Spring System Subject to Its Input Constraints, Proc. Am. Control Conf., Chicago, USA, June 28–30, 2000, pp. 1817–1821.Google Scholar
  12. 12.
    Polyak, B.T., Convexity of Quadratic Transformations and Its Use in Control and Optimization, J. Optim. Theory Appl., 1998, vol. 99, pp. 553–583.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2012

Authors and Affiliations

  • M. V. Khlebnikov
    • 1
  1. 1.Trapeznikov Institute of Control SciencesRussian Academy of SciencesMoscowRussia

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