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Automation and Remote Control

, Volume 79, Issue 7, pp 1222–1239 | Cite as

Quadratic Stabilization of Discrete-Time Bilinear Systems

  • M. V. Khlebnikov
Nonlinear Systems

Abstract

We consider the problem of stabilization of discrete-time bilinear control systems. Using the linear matrix inequality technique and quadratic Lyapunov functions, we formulate a method for the construction of the so-called stabilizability ellipsoid having the property that the trajectories of the closed-loop system emanating from the points in the ellipsoid asymptotically tend to the origin. The proposed approach allows for an efficient construction of nonconvex domains of stabilizability of discrete-time bilinear control systems. The results are extended to the robust statement of the problem where the system matrix is subjected to structured uncertainties.

Keywords

discrete-time bilinear system quadratic Lyapunov function linear feedback stabilizability ellipsoid domain of stabilizability linear matrix inequalities 

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References

  1. 1.
    Mohler, R.R., Bilinear Control Processes, New York: Academic, 1973.MATHGoogle Scholar
  2. 2.
    Khalil, H.K., Nonlinear Systems, New Yprk: Prentice Hall, 2002. Translated under the title Nelineinye sistemy, Izhevsk: Regulyarnaya i Khaoticheskaya Dinamika, 2009.Google Scholar
  3. 3.
    Isidori, A., Nonlinear Control Systems, London: Springer-Verlag, 1995.CrossRefMATHGoogle Scholar
  4. 4.
    Krstić, M., Kanellakopoulos, I., and Kokotovic, P., Nonlinear and Adaptive Control Design, New York: Wiley, 1995.MATHGoogle Scholar
  5. 5.
    Emel’yanov, S.V. and Krishchenko, A.P., Stabilizability of Bilinear Systems of Canonical Form, Dokl. Math., 2012, vol. 86, no 1, pp. 591–594.MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Emel’yanov, S.V, Korovin, S.K., and Shepit’ko, A.S., Stabilization of Bilinear Systems on the Plane by Constant and Relay Controls, Differ. Equat., 2000, vol. 36, no 8, pp. 1131–1138.MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Ryan, E. and Buckingham, N., On Asymptotically Stabilizing Feedback Control of Bilinear Systems, IEEE Trans. Autom. Control, 1983, vol. 28, no. 8, pp. 863–864.CrossRefMATHGoogle Scholar
  8. 8.
    Chen, L.K., Yang, X., and Mohler, R.R., Stability Analysis of Bilinear Systems, IEEE Trans. Autom. Control, 1991, vol. 36, no. 11, pp. 1310–1315.MathSciNetCrossRefGoogle Scholar
  9. 9.
    Fomichev, V.V. and Shepit’ko, A.S., The Method of Rotating Lyapunov Functions in the Stabilization Problem for Two-dimensional Bilinear Systems, Differ. Equat., 2000, vol. 36, no 8, pp. 1262–1265.CrossRefMATHGoogle Scholar
  10. 10.
    Hu, B., Zhai, G., and Michel, A.N., Stabilization of Two-Dimensional Single-Input Bilinear Systems with a Finite Number of Constant Feedback Controllers, Proc. Am. Control Conf., Anchorage, USA, May 2002, vol. 3, pp. 1874–1879.Google Scholar
  11. 11.
    Čelikovský, S., On the Global Linearization of Bilinear Systems, Syst. Control Lett., 1990, vol. 15, no. 5, pp. 433–439.MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Čelikovský, S., On the Stabilization of the Homogeneous Bilinear Systems, Syst. Control Lett., 1993, vol. 21, no. 6, pp. 503–510.MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Tibken, B., Hofer, E.P., and Sigmund, A., The Ellipsoid Method for Systematic Bilinear Observer Design, Proc. 13th IFAC World Congr., San Francisco, USA, June–July 1996, pp. 377–382.Google Scholar
  14. 14.
    Korovin, S.K. and Fomichev, V.V., Asymptotic Observers for Some Classes of Bilinear Systems with Linear Input, Dokl. Math., Control Theory, 2004, vol. 398, no 1, pp. 38–43.Google Scholar
  15. 15.
    Belozyorov, V.Y., Design of Linear Feedback for Bilinear Control Systems, Int. J. Appl. Math. Comput. Sci., 2002, vol. 11, no. 2, pp. 493–511.MathSciNetMATHGoogle Scholar
  16. 16.
    Belozyorov, V.Y., On Stability Cones for Quadratic Systems of Differential Equations, J. Dynam. Control Syst., 2005, vol. 11, no. 3, pp. 329–351.MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Amato, F., Cosentino, C., and Merola, A., Stabilization of Bilinear Systems via Linear State Feedback Control, IEEE Trans. Circuits Syst. II. Express Briefs, 2009, vol. 56, no. 1, pp. 76–80.CrossRefGoogle Scholar
  18. 18.
    Andrieu, V. and Tarbouriech, S., Global Asymptotic Stabilization for a Class of Bilinear Systems by Hybrid Output Feedback, IEEE Trans. Autom. Control, 2013, vol. 58, no. 6, pp. 1602–1608.MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Coutinho, D., and de Souza, C.E., Nonlinear State Feedback Design with a Guaranteed Stability Domain for Locally Stabilizable Unstable Quadratic Systems, IEEE Trans. Circuits Syst. I. Regular Papers, 2012, vol. 59, no. 2, pp. 360–370.MathSciNetCrossRefGoogle Scholar
  20. 20.
    Omran, H., Hetel, L., Richard, J.-P., et al., Stability Analysis of Bilinear Systems under Aperiodic Sampled-Data Control, Automatica, 2014, vol. 50, no. 4, pp. 1288–1295.MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Kung, C.-C., Chen, T.-H., Chen, W.-C., et al., Quasi-Sliding Mode Control for a Class of Multivariable Discrete Time Bilinear Systems, Proc. 2012 IEEE Int. Conf. Syst., Man, Cybernet., Seoul, Korea, October 2012, pp. 1878–1883.CrossRefGoogle Scholar
  22. 22.
    Goka, T., Tarn, T.J., and Zaborszky, J., On the Controllability of a Class of Discrete Bilinear Systems Automatica, 1973, vol. 9, no. 5, pp. 615–622.MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Tie, L. and Lin, Y., On Controllability of Two-Dimensional Discrete-Time Bilinear Systems Int. J. Syst. Sci., 2015, vol. 46, no. 10, pp. 1741–1751.MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Athanasopoulos, N. and Bitsoris, G., Constrained Stabilization of Bilinear Discrete-Time Systems Using Polyhedral Lyapunov Functions, Proc. 17th IFAC World Congr., Seoul, Korea, July 6–11, 2008, pp. 2502–2507.Google Scholar
  25. 25.
    Athanasopoulos, N. and Bitsoris, G., Stability Analysis and Control of Bilinear Discrete-Time Systems: A Dual Approach, Proc. 18th IFAC World Congr., Milan, Italy, August 28–September 2, 2011, pp. 6443–6448.Google Scholar
  26. 26.
    Tarbouriech, S., Queinnec, I., Calliero, T.R., et al., Control Design for Bilinear Systems with a Guaranteed Region of Stability: An LMI-Based Approach, Proc. 17th Mediterran. Conf. Control Autom., Thessaloniki, Greece, June 2009, pp. 809–814.Google Scholar
  27. 27.
    Boyd, S., El Ghaoui, L., Feron, E., et al., Linear Matrix Inequalities in System and Control Theory, Philadelphia: SIAM, 1994.CrossRefMATHGoogle Scholar
  28. 28.
    Khlebnikov, M.V., Quadratic Stabilization of Bilinear Control Systems, Autom. Remote Control, 2016, vol. 77, no. 6, pp. 980–991.MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Petersen, I.R., A Stabilization Algorithm for a Class of Uncertain Linear Systems, Syst. Control Lett., 1987, vol. 8, no. 4, pp. 351–357.MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Khlebnikov, M.V. and Shcherbakov, P.S., Petersen’s Lemma on Matrix Uncertainty and Its Generalization, Autom. Remote Control, 2008, vol. 69, no. 11, pp. 1932–1945.MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Khlebnikov, M.V., New Generalizations of the Petersen Lemma, Autom. Remote Control, 2014, vol. 75, no. 5, pp. 917–921.MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Polyak, B.T., Khlebnikov, M.V, and Shcherbakov, P.S., Upravlenie lineinymi sistemami pri vneshnikh vozmushcheniyakh: tekhnika lineinykh matrichnykh neravenstv (Control of Linear Systems Subject to Exogenous Disturbances: The Linear Matrix Inequalitiy Technique), Moscow: LENAND, 2014.Google Scholar
  33. 33.
    Horn, R. and Johnson, Ch., Matrix Analysis, New York: Cambridge Univ. Press, 2012, 2nd ed. Translated under the title Matrichnyi analiz, Moscow: Mir, 1989.Google Scholar
  34. 34.
    Grant, M., and Boyd, S., CVX: Matlab Software for Disciplined Convex Programming, version 2.0 beta. http://cvxr.com/cvx, September 2013.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

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

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