Stability of LTI Systems with Unstructured Uncertainty Using Quadratic Disc Criterion

  • Yeom, Dong-Hae (BK21 Nourishment of Human Resource for Embedded Systems, Kunsan National University) ;
  • Park, Jin-Bae (Dept. of Electrical and Electronics Engineering, Yonsei University) ;
  • Joo, Young-Hoon (Dept. of Control and Robotics Engineering, Kunsan National University)
  • Received : 2010.11.25
  • Accepted : 2011.06.24
  • Published : 2012.01.01


This paper deals with robust stability of linear time-invariant (LTI) systems with unstructured uncertainties. A new relation between uncertainties and system poles perturbed by the uncertainties is derived from a graphical analysis. A stability criterion for LTI systems with uncertainties is proposed based on this result. The migration range of the poles in the proposed criterion is represented as the bound of uncertainties, the condition number of a system matrix, and the disc containing the poles of a given nominal system. Unlike the existing methods depending on the solutions of algebraic matrix equations, the proposed criterion provides a simpler way which does not involves algebraic matrix equations, and a more flexible root clustering approach by means of adjusting the center and the radius of the disc as well as the condition number.


Supported by : National Research Foundation of Korea


  1. J. Bosche, O. Bachelier, D. Mehdi, "An Approach for Robust Matrix Root-clustering Analysis in a Union of Regions," IMA Journal of Mathematical Control and Information, Vol. 22, No. 3, pp. 227-239, 2005.
  2. R. D. Braatz and M. Morari, "Minimizing the Euclidean Condition Number," SIAM Journal of Control and Optimization, Vol. 32, No. 6, pp. 1763-1768, Nov. 1994.
  3. C. H. Fang and C. L. Lu, "Robustness of Pole-Clustering in a Ring for Structured Perturbation Systems," IEEE Transactions on Circuits and Systems, Vol. 43, No. 6, pp. 496-500, June 1996.
  4. M. E. Halpem, R. J. Evans, and R. D. Hill, "Pole Assignment with Robust Stability," IEEE Transactions on Automatic Control, Vol. 40, No. 4, pp. 725-729, Apr. 1995.
  5. D. Hinrichsen and A. J. Pritchard, "Robustness Measures for Linear Systems with Application to Stability Radii of Hurwitz and Schur Polynomials," International Journal of Control, Vol. 55, No. 4 pp. 809-844, 1992.
  6. Z. Gajic, Lyapunov Matrix Equation in System Stability and Control, Academic Press, pp. 245-246, 1995.
  7. G. Garcia and J. Bemussou, "Pole Assignment for Uncertain Systems in a Specified Disk by State Feedback," IEEE Transactions on Automatic Control, Vol. 40, No. 1, pp. 184-190, Jan. 1995.
  8. C. Greif and J. M. Varah, "Minimizing the Condition Number for Small Rank Modifications," SIAM Journal of Matrix Analysis and Applications, Vol. 29, No. 1, pp. 82-97, 2006.
  9. J. Kautsky and N. K. Nichols and P. Van Dooren, "Robust Pole Assignment in Linear State Feedback," International Journal of Control, Vol. 41, No. 5, pp. 1129-1155, 1985.
  10. P. Lancaster and M. Tismenetsky, The Theory of matrices, Academic Press, pp.387-390, 1985.
  11. D. Peaucelle, D. Arzelier, O.L Bachelier, J. Bernussou, "A New Robust D-stability Condition for Real Convex Polytopic Uncertainty," Systems and Control Letters, Vol. 40, No. 1, pp. 21-30, May 2000.
  12. M. A. Rami, S. E. Faiz, A. Benzaouia, and F. Tadeo, "Robust Exact Pole Placement via an LMI-Based Algorithm," IEEE Transactions on Automatic Control, Vol. 54, No. 2, pp. 394-398, Feb. 2009.
  13. M. T. Soylemez and N. Munro, "Robust Pole Assignment in Uncertain Systems," IEE Proceedings of Control Theory and Applications, Vol. 144, No. 3, pp. 217-224, May 1997.
  14. A. Varga, "Robust and Minimum Norm Pole Assignment with Periodic State Feedback," IEEE Transactions on Automatic Control, Vol. 45, No. 5, pp. 1017-1022, May 2000.
  15. R. S. Varga, Gerschgorin and His Circles, Springer-Verlag, Berlin Heidelberg, 2004.
  16. D. H. Yeom, K. H. Im, and J. Y. Choi, "New Stability Criterion and Pole Assignment for Switched Linear Systems," International Journal of Control, Automation, and Systems, Vol. 3, No. 4, pp. 580-590, Dec. 2005.