Finite time control for one class of nonlinear systems with the H performance criterion

Control in Stochastic Systems and Under Uncertainty Conditions


This work considers nonautonomous control systems with uncertain nonlinearities subjected to the impact of external disturbances which are continuous functions bounded by the L 2-norm. Based on the method of matrix comparison systems and technique of differential linear matrix inequalities, it is proposed to solve problems of finite time stability and boundedness with respect to the given sets, as well as suppress the disturbances and initial deviations using the state feedback with an estimation of the performance by the H -criterion.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    F. C. Schweppe, Uncertain Dynamic Systems (Prentice Hall, Upper Saddle River, NJ, 1973).Google Scholar
  2. 2.
    F. L. Chernous’ko, Estimation of the Phase State of Dynamical Systems. The Method of Ellipsoids (Nauka, Moscow, 1988) [in Russian].MATHGoogle Scholar
  3. 3.
    A. V. Kurzhanskii and I. Valyi, Ellipsoidal Calculus for Estimation and Control (Birkhauser, Boston, 1997).CrossRefMATHGoogle Scholar
  4. 4.
    S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory (SIAM, Philadelphia, 1994).CrossRefMATHGoogle Scholar
  5. 5.
    D. V. Balandin and M. M. Kogan, Synthesis of Control Laws Based on Linear Matrix Inequalities (Fizmatlit, Moscow, 2007) [in Russian].MATHGoogle Scholar
  6. 6.
    S. A. Nazin, B. T. Polyak, and M. V. Topunov, “Rejection of bounded exogenous disturbances by the method of invariant ellipsoids,” Autom. Remote Control 68, 467–486 (2007).MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    B. T. Polyak, M. V. Topunov, and P. S. Shcherbakov, “Ideology of invariant ellipsoid in the problem on robust suppression of bounded exogenous disturbances,” in Stochastical Optimization in Informatics, Ed. by O. N. Granichin (St.Pb. Gos. Univ., St. Petersburg, 2007), No. 3, pp. 51–84.Google Scholar
  8. 8.
    B. T. Polyak and M. V. Topunov, “Suppression of bounded exogenous disturbances: output feedback,” Autom. Remote Control 69, 801 (2008).MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    M. V. Khlebnikov, “Suppression of bounded exogenous disturbances: A linear dynamic output controller,” Autom. Remote Control 72, 699–712 (2011).MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    D. D. Siljak and D. M. Stipanovic, “Robust stabilization of nonlinear systems: the LMI approach,” Math. Probl. Eng. 6, 461–493 (2000).MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    A. I. Zecevic and D. D. Siljak, Control of Complex Systems. Structural Constraints and Uncertainty, Communications and Control Engineering Series (Springer Science, New York, 2010).MATHGoogle Scholar
  12. 12.
    B. T. Polyak, M. V. Khlebnikov, and P. S. Shcherbakov, “Nonlinear systems with bounded or multiplicative disturbances,” in Problems of Stability and Control, Collection of Articles Dedicated to 80 Years from Acad. V. M. Matrosov’ Birthday (Fizmatlit, Moscow, 2013), pp. 270–299 [in Russian].Google Scholar
  13. 13.
    D. V. Balandin and M. M. Kogan, “Synthesis of optimal robust H -control by convex optimization methods,” Autom. Remote Control 65, 1099–1110 (2004).MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    D. V. Balandin and M. M. Kogan, “Linear matrix inequalities in the robust output H control problem,” Dokl. Math. 69, 488 (2004).MATHGoogle Scholar
  15. 15.
    D. V. Balandin and M. M. Kogan, “An optimization algorithm for checking feasibility of robust H -control problem for linear time-varying uncertain systems,” Int. J. Control. 77, 498–503 (2004).MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    D. V. Balandin, M. M. Kogan, “Linear-quadratic and γ-optimal output control laws,” Autom. Remote Control 69, 911–919 (2008).MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    D. V. Balandin and M. M. Kogan, “LMI-based H -optimal control with transient,” Int. J. Control. 83, 1664–1673 (2010).MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    D. V. Balandin and M. M. Kogan, “Generalized H -optimal control as a trade-off between the H -optimal and γ-optimal controls,” Autom. Remote Control 71, 993–1010 (2010).MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    N. D. Moiseev, Survey of Stability Theory Development (Gostekhizdat, Moscow, Leningrad, 1949) [in Russian].Google Scholar
  20. 20.
    G. V. Kamenkov, “On stability of motion over a finite interval of time,” Prikl. Mat. Mekh. 17, 529–540 (1953).MathSciNetGoogle Scholar
  21. 21.
    A. A. Lebedev, “On stability of motion over a specified interval of time,” Prikl. Mat. Mekh. 18, 139–148 (1954).Google Scholar
  22. 22.
    F. Amato, M. Ariola, and P. Dorato, “Finite-time control of linear systems subject to parametric uncertainties and disturbances,” Automatica 37 (9), 145–146 (2001).CrossRefMATHGoogle Scholar
  23. 23.
    F. Amato, M. Ariola, and C. Cosentino, “Finite-time stabilization via dynamic output feedback,” Automatica 42, 337–342 (2006).MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    F. Amato, M. Ariola, and C. Cosentino, “Finite-time control of linear time-varying systems via output feedback,” in Proceedings of the American Control Conference, Portland Hilton, Oregon, June 8–10, 2005 (IEEE, New York, 2005), pp. 4722–4726.Google Scholar
  25. 25.
    F. Amato, R. Ambrosino, M. Ariola, C. Cosentino, and G. de Tommasi, Finite-Time Stability and Control (Springer, London, 2014).CrossRefMATHGoogle Scholar
  26. 26.
    P. P. Khargonekar, K. M. Nagpal, and K. R. Poola, “Control with transients,” SIAM J. Control 29, 1373–1393 (1991).MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Y. K. Foo, “Control with initial conditions,” IEEE Trans. Circuits Syst. 53, 867–871 (2006).CrossRefGoogle Scholar
  28. 28.
    Q. Meng and Y. Shen, “Finite-time control for linear continuous system with norm-bounded disturbance,” Commun. Nonlin. Sci. Numer. Simul. 14, 1043–1049 (2009).MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Y. Guo, Y. Yao, S. Wang, B. Yang, K. Liu, and X. Zhao, “Finite-time control with H-infinity constraints of linear time-invariant and time-varying systems,” J. Control Theory Appl. 11, 165–172 (2013).MathSciNetCrossRefGoogle Scholar
  30. 30.
    N. S. Postnikov and E. F. Sabaev, “Matrix comparison systems and their applications to automatic control problems,” Autom. Remote Control 41, 455–464 (1980).Google Scholar
  31. 31.
    A. I. Malikov and A. E. Blagov, “Dynamical analysis of control systems using matrix comparison systems,” Vestn. Kazan. Tekh. Univ., No. 4, 71–75 (1996).Google Scholar
  32. 32.
    A. I. Malikov and A. E. Blagov, “Analysis of dynamics of multiply-connected control systems using matrix comparison systems,” Vestn. Kazan. Tekh. Univ., No. 2, 37–43 (1998).Google Scholar
  33. 33.
    A. I. Malikov, “Matrix comparison systems in the analysis of dynamics of control systems with structural changes,” J. Comput. Syst. Sci. Int. 38, 343 (1999).MATHGoogle Scholar
  34. 34.
    A. I. Malikov, “Matrix systems of differential equations with the quasimonotonicity condition,” Russ. Math. (Izv. VUZ) 44 (8), 33–43 (2000).MathSciNetMATHGoogle Scholar
  35. 35.
    A. I. Malikov, “Algorithms for dynamic and performance analysis of control systems with uncertainties based on matrix comparison systems,” in Proceedings of the 2nd Russian Conference with International Participation on Technical Means and Software of Control and Measurement Systems (Theory, Methods, Algorithms, Studies and Developments) (Inst. Prikl. Upravl. RAN, Moscow, 2010), CD-ROM, pp. 836–842.Google Scholar
  36. 36.
    S. N. Vassilyev, A. A. Kosov, and A. I. Malikov, “Stability analysis of nonlinear switched systems via reduction method,” IFAC-PapersOnline 18, 5718–5723 (2011).Google Scholar
  37. 37.
    A. I. Malikov, “Method of matrix comparison systems in analysis and synthesis of control systems with uncertainties,” in Proceedings of the 10th International Chetaev Conference on Analytical Mechanics, Stability and Control (Kazan. Gos. Tekh. Univ., Kazan’, 2012), Vol. 2, pp. 360–370.Google Scholar
  38. 38.
    M. N. Elbsat and E. E. Yaz, “Robust and resilient finite-time control of a class of continuous-time nonlinear systems,” in Proceedings of the 7th IFAC Symposium on Robust Control Design ROCOND 2012, Aalborg, Denmark, June 20–22, 2012 (IFAC, 2012), pp. 15–20.Google Scholar
  39. 39.
    F. Feng, C. S. Jeong, E. E. Yaz, S. C. Schneider, and Y. I. Yaz, “Robust controller design with general criteria for uncertain conic nonlinear systems with disturbances,” in Proceedings of the American Control Conference, Washington DC, June 17–19, 2013, pp. 5869–5874.Google Scholar
  40. 40.
    E. A. Fedosov, V. V. Insarov, and O. S. Selivokhin, Control Systems for Object Terminal Positioning under Enviroment’s Counteraction (Nauka, Moscow, 1989) [in Russian].MATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Kazan National Research Technical University Named after A.N. Tupolev, Institute of Mechanics and Engineering, Kazan Science CenterRussian Academy of SciencesKazan, TatarstanRussia

Personalised recommendations