Abstract
This technical note investigates the problem of checking robust D-stability of polytopes of polynomial matrices. Lifted linear matrix inequality (LMI) conditions with two-DOF (two degree of freedom) positive integers (τ, κ) are derived to possess more flexible tradeoff between the conservatism and computational complexity. In the process of formulating the LMIs, the relevant region D is represented by a quadratic constraint in the complex plane. The matrix, composing the quadratic form with the vector of a variable, is called the region matrix. Then a variable substitution approach is put forward for the lifted LMI version by extending the dimensions of the region matrix and the Lyapunov matrix. The effectiveness and advantages of the proposed method have been illustrated by numerical examples.
Article PDF
Avoid common mistakes on your manuscript.
References
W. C. Karl and G. C. Verghese, “A sufficient condition for the stability of interval matrix polynomials,” IEEE Trans. on Automatic Control, vol. 38, no. 7, pp. 1139–1143, July 1993.
D. Henrion, D. Arzelier, D. Peaucelle, and M. Sebek, “An LMI condition for robust stability of polynomial matrix polytopes,” Automatica, vol. 37, no. 3, pp. 461–468, March 2001.
L. Gurvits and A. Olshevsky, “On the NP-hardness of checking matrix polytope stability and continuous-time switching stability,” IEEE Trans. on Automatic Control, vol. 54, no. 2, pp. 337–341, February 2009.
P. P. Khargonekar, I. R. Petersen, and K. M. Zhou, “Robust Stabilization of Uncertain Linear Systems Quadratic Stabilizability and H ∞ Control Theory,” IEEE Trans. on Automatic Control, vol. 35, no. 3, pp. 356–361, March 1990.
M. Chilali and P. Gahinet, “H ∞ design with pole placement constraints: an LMI approach,” IEEE Trans. on Automatic Control, vol. 41, no. 3, pp. 358–367, March 1996.
D. Henrion, O. Bachelier, and M. Sebek, “— D-stability of polynomial matrices,” International Journal of Control, vol. 74, no. 8, pp. 845–856, August 2001.
C. W. Scherer, “LMI relaxations in robust control,” European Journal of Control Engineering Practice, vol. 12, no. 1, pp. 3–30, January 2006.
J. G. Lu and Y. Q. Chen, “Robust stability and sta bilization of fractional-order interval systems with the fractional order alpha: the 0 < α < 1 case,” IEEE Trans. on Automatic Control, vol. 55, no. 1, pp. 152–158, January 2010.
V. J. S. Leite and P. L. D. Peres, “An improved LMI condition for robust — D-stability of uncertain polytopic systems,” IEEE Trans. on Automatic Control, vol. 48, no. 3, pp. 500–504, March 2003.
R. C. L. F. Oliveira and P. L. D. Peres, “Parameterdependent LMIs in robust analysis: characterization of homogeneous polynomially parameter-dependent solutions via LMI relaxations,” IEEE Trans. on Automatic Control, vol. 52, no. 7, pp. 1334–1340, July 2007.
R. C. L. F. Oliveira, M. C. de Oliveira, and P. L. D. Peres, “Convergent LMI relaxations for robust analysis of uncertain linear systems using lifted polynomial parameter-dependent Lyapunov functions,” Systems and Control Letters, vol. 57, no. 8, pp. 680–689, August 2008.
Y. Ebigara, K. Maeda, and T. Hagiwara, “Robust — D-stability analysis of uncertain polynomial matrices via polynomial-type multipliers,” Proc. 16th IFAC World Congress, Czech Republic, vol. 16, part 1, pp. 975–980, 2005.
T. M. Guerra, A. Kruszewski, and M. Bernal, “Control law proposition for the stabilization of discrete Takagi-Sugeno models,” IEEE Trans. on Fuzzy Systems, vol. 17, no. 3, pp. 724–731, March 2009.
D. H. Lee, J. B. Park, and Y. H. Joo, “A less conservative LMI condition for robust — D-stability of polynomial matrix polytopes — a projection approach,” IEEE Trans. on Automatic Control, vol. 56, no. 4, pp. 868–873, April 2011.
D. H. Lee, J. B. Park, Y. H. Joo, and K. C. Lin, “Lifted versions of robust — D-stability and — D-stabilisation conditions for uncertain polytopic linear systems,” IET Control Theory and Applications, vol. 6, no. 1, pp. 24–36, January 2012.
I. Gohberg, P. Lancaster, and L. Rodman, Matrix Polynomials, Academic, New York, 1982.
S. Gutman and E. I. Jury, “General theory for matrix root-clustering in subregions of the complex plane,” IEEE Trans. on Automatic Control, vol. 26, no. 4, pp. 853–863, April 1981.
R. E. Skelton, T. Iwasaki, and K. Grigoriadis, A Unified Algebraic Approach to Linear Control Design, Taylor and Francis, New York, 1998.
A. Benzaouia, S. Gounane, F. Tadeo, and A. El Hajjaji, “Stabilization of saturated discrete-time fuzzy systems,” International Journal of Control, Automation and Systems, vol. 9, no. 3, pp. 581–587, June 2011.
X. H. Chang, “H ∞ Controller Design for Linear Systems with Time-invariant Uncertainties,” International Journal of Control, Automation and Systems,, vol. 9, no. 2, pp. 391–395, April 2011.
M. M. Belhaouane and N. B. Braiek, “Design of stabilizing control for synchronous machines via polynomial modelling and linear matrix inequalities approach,” International Journal of Control, Automation and Systems, vol. 9, no. 3, pp. 425–436, June 2011.
Author information
Authors and Affiliations
Corresponding author
Additional information
Recommended by Editorial Board member Young Soo Suh under the direction of Editor Ju Hyun Park.
This work was supported by the National Natural Science Foundation of China (Grant Nos. 61004017 and 60974103) and the National 863 Project of China (Grant No. 2011AA7034056).
Yong Wang received his Ph.D. degree in Automation from Nanjing University of Aeronautics and Astronautics. He is currently a Professor and Supervisor in University of Science and Technology of China. His research interests include robust control, fractional order systems, active vibration control.
Shu Liang received his B.Eng. degree in Automation from University of Science and Technology of China in 2010 and currently is a Ph.D. candidate of grade one. His research interests include robust control, fractional order systems control and approximation.
Rights and permissions
About this article
Cite this article
Wang, Y., Liang, S. Two-DOF lifted LMI conditions for robust D-stability of polynomial matrix polytopes. Int. J. Control Autom. Syst. 11, 636–642 (2013). https://doi.org/10.1007/s12555-012-0471-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12555-012-0471-9