Skip to main content
SpringerLink
Log in

Robust Stabilization of Discrete-Time Uncertain Nonlinear Systems

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

This paper applies a recent result on nonlinear H ∞-optimal control to give a necessary and sufficient condition for the existence of a state feedback controller which absolutely stabilizes a discrete-time uncertain nonlinear system with structured uncertainty. The class of uncertain systems being considered contain uncertainties which satisfy a certain sum constraint. This sum constraint is a nonlinear discrete-time version of the integral quadratic constraint uncertainty description. The main result is established by using a nonlinear S-procedure result, which enables the robust absolute stabilization problem to be converted into a equivalent nonlinear H∞-optimal control problem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. James, M., and Baras, J., Robust H ∞-Output Feedback Control for Nonlinear Systems, IEEE Transactions on Automatic Control, Vol. 40, pp. 1007–1017, 1995.

    Google Scholar 

  2. Isidori, A., and Astolfi, A., Disturbance Attenuation and H ∞-Control via Measurement Feedback, IEEE Transactions on Automatic Control, Vol. 37, pp. 1283–1293, 1992.

    Google Scholar 

  3. Lu, W. M., and Doyle, J. C., Robustness Analysis and Synthesis for Uncertain Nonlinear Systems, Proceedings of the 33rd IEEE Conference on Decision and Control, Lake Buena Vista, Florida, pp. 787–792, 1994.

  4. Yakubovich, V. A., Minimization of Quadratic Functionals under Quadratic Constraints and Necessity of a Frequency Condition in the Quadratic Criterion for Absolute Stability of Nonlinear Control Systems, Soviet Mathematics Doklady, Vol. 14, pp. 593–597, 1973.

    Google Scholar 

  5. Yakubovich, V. A., Abstract Theory of Absolute Stability of Nonlinear Systems, Vestnik Leningrad University, Series 1, Vol. 41, pp. 99–118, 1977.

  6. Yakubovich, V. A., Dichotomy and Absolute Stability of Nonlinear Systems with Periodically Nonstationary Linear Part, Systems and Control Letters, Vol. 11, pp. 221–228, 1988.

    Google Scholar 

  7. Yakubovich, V. A., Absolute Stability of Nonlinear Systems with a Periodically Nonstationary Linear Part, Soviet Physics Doklady, Vol. 32, pp. 5–7, 1988.

    Google Scholar 

  8. Savkin, A. V., and Petersen, I. R., A Connection between H ∞-Control and the Absolute Stabilizability of Uncertain Systems, Systems and Control Letters, Vol. 23, pp. 197–203, 1994.

    Google Scholar 

  9. Savkin, A. V., and Petersen, I. R., Nonlinear versus Linear Control in the Absolute Stabilizability of Uncertain Linear Systems with Structured Uncertainty, IEEE Transactions on Automatic Control, Vol. 40, pp. 122–127, 1995.

    Google Scholar 

  10. Savkin, A. V., and Petersen, I. R., Minimax Optimal Control of Uncertain Systems with Structured Uncertainty, International Journal of Robust and Nonlinear Control, Vol. 5, pp. 119–137, 1995.

    Google Scholar 

  11. Doyle, J. C., Analysis of Feedback Systems with Structured Uncertainty, IEE Proceedings, Vol. 129D, pp. 242–250, 1982.

    Google Scholar 

  12. Yakubovich, V. A., S-Procedure in Nonlinear Control Theory, Vestnik Leningrad University, Series 1, Vol. 13, pp. 62–77, 1971.

  13. Uhlig, F., A Recurring Theorem about Pairs of Quadratic Forms and Extensions: A Survey, Linear Algebra and Its Applications, Vol. 25, pp. 219–237, 1979.

    Google Scholar 

  14. Megretsky, A., Necessary and Sufficient Conditions of Stability: A Multiloop Generalization of the Circle Criterion, IEEE Transactions on Automatic Control, Vol. 38, pp. 753–756, 1993.

    Google Scholar 

  15. Megretsky, A., and Treil, S., Power Distribution Inequalities in Optimization and Robustness of Uncertain Systems, Journal of Mathematical Systems, Estimation, and Control, Vol. 3, pp. 301–319, 1993.

    Google Scholar 

  16. Willems, J. C., Dissipative Dynamical Systems, Part 1: General Theory, Archive for Rational Mechanics and Analysis, Vol. 45, pp. 321–351, 1972.

    Google Scholar 

  17. Hill, D., and Moylan, P., The Stability of Nonlinear Dissipative Systems, IEEE Transactions on Automatic Control, Vol. 21, pp. 708–7811, 1976.

    Google Scholar 

  18. Hill, D., and Moylan, P., Stability Results for Nonlinear Feedback Systems, Automatica, Vol. 13, pp. 377–382, 1977.

    Google Scholar 

  19. James, M. R., A Partial Differential Inequality for Dissipative Nonlinear Systems, Systems and Control Letters, Vol. 21, pp. 315–320, 1993.

    Google Scholar 

  20. Savkin, A. V., and Petersen, I. R., Nonlinear versus Linear Control in the Absolute Stabilizability of Uncertain Linear Systems with Structured Uncertainty, Proceedings of the 32nd IEEE Conference on Decision and Control, San Antonio, Texas, pp. 172–177, 1993.

Download references

Author information

Authors and Affiliations

  1. Department of Electrical and Electronic Engineering, University of Western Australia, Nedlands, Perth, Western Australia, Australia

    A. V. Savkin (Senior Lecturer)

  2. School of Electrical Engineering, Australian Defence Force Academy, Canberra, Australia

    I. R. Petersen (Associate Professor)

Authors
  1. A. V. Savkin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. I. R. Petersen
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Savkin, A.V., Petersen, I.R. Robust Stabilization of Discrete-Time Uncertain Nonlinear Systems. Journal of Optimization Theory and Applications 96, 87–107 (1998). https://doi.org/10.1023/A:1022611116645

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1022611116645

Search

Navigation