Automation and Remote Control

, Volume 79, Issue 7, pp 1311–1318 | Cite as

Solving Analysis Problem with Input and Output Disturbances

  • K. O. ZheleznovEmail author
Large Scale Systems Control


A method for solving the analysis problem of a linear control system with input and output disturbances is suggested. Its higher efficiency in comparison with the conventional approach is shown using an example of a test problem from COMPleib.


linear matrix inequalities method of invariant ellipsoids tracking problem 


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  1. 1.
    Akhobadze, A.G. and Krasnova, S.A., Tracking in Linear MIMO Systems under External Perturbations, Autom. Remote Control, 2009, vol. 70, no. 6, pp. 933–957.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Zheleznov, K.O. and Khlebnikov, M.V., Applying the Method of Invariant Ellipsoids for Solving Linear Tracking Problem, Tr. MFTI, 2013, vol. 5, no. 1, pp. 115–121.Google Scholar
  3. 3.
    Lezina, Z.M. and Lezin, V.I., G.V. Shchipanov i teoriya invariantnosti (G.V. Shchipanov and Invariance Theory), Moscow: URSS, 2004.Google Scholar
  4. 4.
    Matrosov, V.M., On the Theory of Stability of Motion, J. Appl. Math. Mech., 1962, vol. 26, no. 6, pp. 1506–1522.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    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.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Polyak, B.T., Khlebnikov, M,V., and Shcherbakov, P.S., Upravlenie lineinymi sistemami pri vneshnikh vozmushcheniyakh: tekhnika lineinykh matrichnykh neravenstv (Control of Linear Systems under Exogenous Disturbances: The Linear Matrix Inequality Technique), Moscow: URSS, 2014.Google Scholar
  7. 7.
    Polyak, B.T. and Shcherbakov, P.S., Robastnaya ustoichivost’ i upravlenie (Robust Stability and Control), Moscow: Nauka, 2002.Google Scholar
  8. 8.
    Sokolov, V.F., Robust Tracking with Unknown Upper Bounds on the Perturbations and Measurement Noise, Autom. Remote Control, 2013, vol. 74, no. 1, pp. 76–89.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Tsykunov, A.M., Robust Tracking System with Compensation of Perturbations and Noises, Vestn. Astrakh. Gos. Tekh. Univ., Ser.: Upravl., Vychisl. Tekhn. Informat., 2014, no. 1, pp. 54–61.Google Scholar
  10. 10.
    Sholokhov, O.V., Minimum-volume Ellipsoidal Approximation of the Sum of Two Ellipsoids, Cybern. Syst. Analysis, 2011, vol. 47, no. 6, pp. 954–960.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Shchipanov, G.V., Theory and Methods of Automatic Controller Design, Avtom. Telemekh., 1939, no. 1, pp. 49–66.Google Scholar
  12. 12.
    Bellman, R., Vector Lyapunov Functions, SIAM J. Control, 1966, vol. 1, no. 1, pp. 32–34.zbMATHGoogle Scholar
  13. 13.
    Bououden, S., Chadli, M., Filali, S., and El Hajjaji, A., Fuzzy Model Based Multivariable Predictive Control of a Variable Speed Wind Turbine: LMI Approach, Renewable Energy, 2012, vol. 37, no. 1, pp. 434–439.CrossRefGoogle Scholar
  14. 14.
    Boyd, S., El Ghaoui, L., Feron, E., and Balakrishan, V., Linear Matrix Inequalities in System and Control Theory, Philadelphia: SIAM, 1994.CrossRefzbMATHGoogle Scholar
  15. 15.
    Cervantes, I. and Alvarez-Ramirez, J., On the PID Tracking Control of Robot Manipulators, Syst. Control Let., 2001, vol. 142, no. 1, pp. 37–46.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Grujić, L.T., Tracking Control of Linear Systems, Boca Raton: CRC Press, 2013.Google Scholar
  17. 17.
    Grujić, L.T. and Mounfield, W.P., Natural Tracking PID Process Control for Exponential Tracking, AIChE J., 1992, vol. 38, no. 4, pp. 555–562.MathSciNetCrossRefGoogle Scholar
  18. 18.
    Kalman, R.E., Contributions to the Theory of Optimal Control, Bol. Soc. Mat. Mexicana, 1960, vol. 5, no. 2, pp. 102–119.MathSciNetGoogle Scholar
  19. 19.
    Leibfritz, F., COMPleib: COnstrained Matrix-optimization Problem library. Version 1.1, Univ. Trier, Germany.
  20. 20.
    Löfberg, J., YALMIP: A Toolbox for Modeling and Optimization in MATLAB, 2004 IEEE Int. Symp. on Computer Aided Control Systems Design, 2004, pp. 284–289.Google Scholar
  21. 21.
    Liao, F., Wang, J.L., and Yang, G.-H., Reliable Robust Flight Tracking Control: An LMI Approach, IEEE Trans. Control Syst. Technol., 2002, vol. 10, no. 1, pp. 76–89.CrossRefGoogle Scholar
  22. 22.
    Phat, V.N., Knogtham, Y., and Ratchagit, K., LMI Approach to Exponential Stability of Linear Systems with Interval Time-Varying Delays, Linear Algebra Appl., 2012, vol. 436, no. 1, pp. 243–251.MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Skogestad, S. and Poslethwaite, I., Multivariable Feedback Control: Analysis and Design, New York: Wiley, 2007.Google Scholar
  24. 24.
    Toh, K., Todd, M., and Tütüncü, R., SDPT3-a MATLAB Software Package for Semidefinite Programming, Version 1.3, Optimiz. Methods Software, 1999, vol. 11, no. 1–4, pp. 545–581.MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Tütüncü, R., Toh, K., and Todd, M., Solving Semidefinite-Quadratic-Linear Programs Using SDPT3, Math. Program., 2003, vol. 95, no. 2, pp. 189–217.MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Zhang, H., Shi, I., Mehr, A.S., and Huang, H., Robust FIR Equalization for Time-Varying Communication Channels with Intermittent Observations via an LMI Approach, Signal Proces., 2011, vol. 91, no. 7, pp. 1651–1658.CrossRefzbMATHGoogle Scholar
  27. 27.
    Zhang, B., Zheng, W.X., and Xu, S., Passivity Analysis and Passive Control of Fuzzy Systems with Time-Varying Delays, Fuzzy Sets Syst., 2011, vol. 174, no. 1, pp. 83–98.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Trapeznikov Institute of Control SciencesRussian Academy of SciencesMoscowRussia

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