Skip to main content
SpringerLink
Log in

LMI Approach to Output Feedback Control for Linear Uncertain Systems with D-Stability Constraints

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

Abstract

This paper deals with the problem of designing output feedback controllers for linear uncertain continuous-time and discrete-time systems with circular pole constraints. The uncertainty is assumed to be norm bounded and enters into both the system state and input matrices. We focus on the design of a dynamic output feedback controller that, for all admissible parameter uncertainties, assigns all the closed-loop poles inside a specified disk. It is shown that the problem addressed can be recast as a convex optimization problem characterized by linear matrix inequalities (LMI); therefore, an LMI approach is developed to derive the necessary and sufficient conditions for the existence of all desired dynamic output feedback controllers that achieve the specified circular pole constraints. An effective design procedure for the expected controllers is also presented. Finally, a numerical example is provided to show the usefulness and applicability of the present approach.

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. FURUTA, K., and KIM, S. B., Pole Assignment in a Specified Disk, IEEE Transactions on Automatic Control, Vol. 32, pp. 423-427, 1987.

    Google Scholar 

  2. HADDAD, W. M., and BERNSTEIN, D. S., Controller Design with Regional Pole Constraints, IEEE Transactions on Automatic Control, Vol. 37, pp. 54-69, 1992.

    Google Scholar 

  3. KAWASAKI, N., and SHIMEMURA, E., Pole Placement in a Specified Region Based on a Linear Quadratic Regulator, International Journal of Control, Vol. 48, pp. 225-240, 1988.

    Google Scholar 

  4. LEE, T. T., and LEE, S. H., Discrete Optimal Control with Eigenvalues inside a Circular Region, IEEE Transactions on Automatic Control, Vol. 31, pp. 958-962, 1986.

    Google Scholar 

  5. WANG, Z., Parametric Circular Eigenvalue Assignment for Descriptor Systems via State Feedback, IMA Journal of Mathematical Control and Information, Vol. 17, pp. 57-66, 2000.

    Google Scholar 

  6. WANG, Z., CHEN, X., and GUO, Z., Controller Design for Continuous Systems with Variance and Circular Pole Constraints, International Journal of Systems Science, Vol. 26, pp. 1249-1256, 1995.

    Google Scholar 

  7. CHOU, J.H., Pole-Assignment Robustness in a Specified Disk, Systems and Control Letters, Vol. 16, pp. 41-44, 1991.

    Google Scholar 

  8. HORNG, H. Y., CHOU, J. H., and HORNG, I. R., Robustness of Eigenvalue Clustering in Various Regions of the Complex Plane for Perturbed Systems, International Journal of Control, Vol. 57, pp. 1469-1483, 1993.

    Google Scholar 

  9. RACHID, A., Robustness of Pole Assignment in a Specified Region for Perturbed Systems, International Journal of Control, Vol. 21, pp. 579-585, 1990.

    Google Scholar 

  10. JUANG, Y. T., and CHEN, K. H., Robust Pole-Assignment of Linear Dynamic Systems, Control Theory and Advanced Technology, Vol. 5, pp. 67-74, 1989.

    Google Scholar 

  11. TSAY, S. C., FONG, I. K., and KUO, T. S., D-Stability Analysis for Discrete Optimal Regulator, Control Theory and Advanced Technology, Vol. 6, pp. 237-246, 1990.

    Google Scholar 

  12. WANG, Z., Robust HS Control for Uncertain Linear Continuous Systems with Circular Pole Assignment via Output Feedback, Systems Science, Vol. 23, pp. 39-54, 1997.

    Google Scholar 

  13. WANG, Z., TANG, G., and CHEN, X., Robust Controller Design for Uncertain Linear Systems with Circular Pole Constraints, International Journal of Control, Vol. 65, pp. 1045-1054, 1996.

    Google Scholar 

  14. YAZ, E., SKELTON, R. E., and GRIGORIADIS, E., Robust Regional Pole Assignment with Output Feedback, Proceedings of IEEE Conference on Decision and Control, pp. 3009-3013, 1993.

  15. YEDAVALLI, R. K., and LIU, Y., H? Control with Regional Stability Constraints, Automatica, Vol. 31, pp. 611-615, 1995.

    Google Scholar 

  16. GARCIA, G., and BERNUSSOU, J., Pole Assignment for Uncertain Systems in a Specified Disk by State Feedback, IEEE Transactions on Automatic Control, Vol. 40, pp. 184-190, 1995.

    Google Scholar 

  17. DAAFOUZ, J., GARCIA, G., and BERNUSSOU, J., Robust Control of a Flexible Robot Arm Using the Quadratic d-Stability Approach, IEEE Transactions on Control Systems Technology, Vol. 6, pp. 524-533, 1998.

    Google Scholar 

  18. REZA MOHEIMANI, S. O., and PETERSEN, I. R., Quadratic Guaranteed Cost Control with Robust Pole Placement in a Disk, IEE Proceedings on Control Theory and Applications, Vol. 143, pp. 37-43, 1996.

    Google Scholar 

  19. BOYD, S., GHAOUI, L. EL, FERON, E., and BALAKRISHNAN, V., Linear Matrix Inequalities in System and Control Theory, Studies in Applied Mathematics, SIAM, Philadelphia, Pennsylvania, Vol. 15, 1994.

    Google Scholar 

  20. NESTEROV, Y., and NEMIROVSKY, A., Interior-Point Polynomial Methods in Convex Programming: Theory and Applications, SIAM, Philadelphia, Pennsylvania, 1993.

    Google Scholar 

  21. GAHINET, P., and APKARIAN, P., A Linear Matrix Inequality Approach to H? Control, International Journal of Robust and Nonlinear Control, Vol. 4, pp. 421-448, 1994.

    Google Scholar 

  22. XIE, S., XIE, L., and DE SOUZA, C. E., Robust Dissipative Control for Linear Systems with Dissipative Uncertainty, International Journal of Control, Vol. 70, pp. 169-191, 1998.

    Google Scholar 

  23. CHILALI, M., and GAHINET, P., H? Design with Pole Placement Constraints: An LMI Approach, IEEE Transactions on Automatic Control, Vol. 41, pp. 358-367, 1996.

    Google Scholar 

  24. GAHINET, P., NEMIROVSKY, A., LAUB, A. J., and CHILALI, M., LMI Control Toolbox for Use with Matlab, The Math Works, Natick, Massachussets, 1995.

    Google Scholar 

  25. XIE, L., and SOH, Y. C., Robust Control of Linear Systems with Generalized Positive Real Uncertainty, Automatica, Vol. 33, pp. 963-967, 1997.

    Google Scholar 

  26. KHARGONEKAR, P. P., PETERSEN, I. R., and ZHOU, K., Robust Stabilization of Uncertain Linear Systems: Quadratic Stabilization and HS Control Theory, IEEE Transactions on Automatic Control, Vol. 35, pp. 356-361, 1990.

    Google Scholar 

  27. XIE, L., and SOH, Y. C., Guaranteed Cost Control of Uncertain Discrete-Time Systems, Control Theory and Advanced Technology, Vol. 10, pp. 1235-1251, 1995.

    Google Scholar 

  28. XIE, L., Output Feedback HS Control of Systems with Parameter Uncertainty, International Journal of Control, Vol. 63, pp. 741-750, 1996.

    Google Scholar 

  29. PACKARD, A., ZHOU, K., PANDEY, P., and BECKER, G., A Collection of Robust Control Problems Leading To LMIs, Proceedings of the 30th IEEE Conferences on Decision and Control, righton, United Kingdom, pp. 1245-1250, 1991.

Download references

Author information

Authors and Affiliations

  1. Control Theory and Applications Centre, School of Mathematical and Information Sciences, Coventry University, Coventry, United Kingdom

    Z. Wang (Staff Member) & K. J. Burnham (Professor)

Authors
  1. Z. Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. J. Burnham
    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

Wang, Z., Burnham, K.J. LMI Approach to Output Feedback Control for Linear Uncertain Systems with D-Stability Constraints. Journal of Optimization Theory and Applications 113, 357–372 (2002). https://doi.org/10.1023/A:1014887110211

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1014887110211

Search

Navigation