Advertisement

Journal of Optimization Theory and Applications

, Volume 131, Issue 1, pp 135–149 | Cite as

Stabilization of Linear Nonautonomous Systems with Norm-Bounded Controls

  • V. N. Phat
  • P. Niamsup
Article

Abstract

In this paper, we study the stabilization problem for a class of linear nonautonomous systems with norm-bounded controls. Using the Lyapunov function technique, we establish simple verifiable stabilizability conditions without solving any Riccati differential equation. Numerical examples are given to illustrate the results.

Keywords

Stabilizability controllability nonautonomous systems Lyapunov functions Lyapunov equations Riccati differential equations 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    BOYD, S., EL GHAOUI, L., FERON, E., and BALAKRISHNAN, V., Linear Matrix Inequalities and Control Theory, SIAM Studies in Applied Mathematics, SIAM, Philadelphia, Pennsylvania, Vol. 15, 1994.Google Scholar
  2. 2.
    CORLESS, M., and LEITMANN, G., Bounded Controllers for Robust Exponential Convergence, Journal of Optimization Theory and Applications, Vol. 76, pp. 1–12, 1993.CrossRefMathSciNetGoogle Scholar
  3. 3.
    GIBSON, J. S., Riccati Equations and Numerical Approximations, SIAM Journal on Control and Optimization, Vol. 21, pp. 95–139, 1983.CrossRefMathSciNetGoogle Scholar
  4. 4.
    NIAMSUP, P., and PHAT, V. N., Asymptotic Stability of Nonlinear Control Systems Described by Differential Equations with Multiple Delays, Electronic Journal of Differential Equations, Vol. 2000, pp. 1–17, 2000.MathSciNetGoogle Scholar
  5. 5.
    PHAT, V. N., Weak Asymptotic Stabilizability of Discrete Inclusions Given by Set-Valued Operators, Journal of Mathematical Analysis and Applications, Vol. 202, pp. 353–369, 1996.CrossRefMathSciNetGoogle Scholar
  6. 6.
    PHAT, V. N., Stabilization of Linear Continuous Time-Varying Systems with State Delays and Norm-Bounded Uncertainties, Electronic Journal of Differential Equations, Vol. 2001, pp. 1–13, 2001.MathSciNetGoogle Scholar
  7. 7.
    BOUNIT, H., and HAMMOURI, H., Bounded Feedback Stabilization and Global Separation Principle of Distributed-Parameter Systems, IEEE Transactions on Automatic Control, Vol. 42, pp. 414–419, 1997.CrossRefMathSciNetGoogle Scholar
  8. 8.
    SLEMROD, M., Feedback Stabilization of a Linear Control System in Hilbert Space with a priori Bounded Control, Mathematical Control and Signal Systems, Vol. 22, pp. 265–285, 1989.MathSciNetGoogle Scholar
  9. 9.
    SUN, Y. J., LIEN, H., and HSIED, J.G., Global Exponential Stabilization for a Class of Uncertain Nonlinear Systems with Control Constraints, IEEE Transactions on Automatic Control, Vol. 43, pp. 67–70, 1998.Google Scholar
  10. 10.
    SUSSMAN, H. J., SONTAG, E. D., and YANG, Y., A General Result on the Stabilization of Linear Systems Using Bounded Controls, IEEE Transactions on Automatic Control, Vol. 39, pp. 2411–2425, 1994.CrossRefGoogle Scholar
  11. 11.
    SOLDATOS, A. G., and CORLESS, M., Stabilizing Uncertain Systems with Bounded Control, Dynamics and Control, Vol. 1, pp. 227–238, 1991.CrossRefMathSciNetGoogle Scholar
  12. 12.
    SOLDATOS, A.G., CORLESS, M., and LEITMANN, G., Stabilizing Uncertain Systems with Bounded Control, Mechanics and Control, Los Angeles, California, pp. 415–428, 1990; see also Lecture Notes in Control and Information Sciences, Springer, Berlin, Germany Vol. 151, 1991.Google Scholar
  13. 13.
    KALMAN, R., Contribution to the Theory of Optimal Control, Boletin de la Sociedad Matematica Mexicana, Vol. 5, pp. 102–119, 1960.MathSciNetGoogle Scholar
  14. 14.
    IKEDA, M., MAEDA, H., and KOMADA, S., Stabilization of Linear Systems, SIAM Journal on Control, Vol. 10, pp. 716–729, 1972.CrossRefGoogle Scholar
  15. 15.
    MARSHALL, A. W., and OLKIN, L., Inequalities: Theory of Majorizations and Its Applications, Academic Press, New York, NY, 1979.Google Scholar
  16. 16.
    LAUB, A. J., Schur Techniques for Solving Riccati Equations, Feedback Control of Linear and Nonlinear Systems Lecture Notes in Control and Information Sciences, Springer, Berlin, Germany, Vol. 39, pp. 165–174, 1982.Google Scholar
  17. 17.
    ABOU-KANDIL, H., FREILING, G., IONESCU, V., and JANK, G., Matrix Riccati Equations in Control and Systems Theory, Birkhauser, Basel, Switzerland, 2003.zbMATHGoogle Scholar
  18. 18.
    THOMAS, W. R., Riccati Differential Equations, Academic Press, New York, NY, 1972.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • V. N. Phat
    • 1
  • P. Niamsup
    • 2
  1. 1.Institute of MathematicsHanoiVietnam
  2. 2.Department of MathematicsChiangmai UniversityChiangmaiThailand

Personalised recommendations