×
Dabboussi, Kamel; Zrida, Jalel

Sufficient dilated LMI conditions for \(H_\infty\) static output feedback robust stabilization of linear continuous-time systems. (English) Zbl 1235.93085

J. Appl. Math. 2012, Article ID 812920, 13 p. (2012).
Summary: New sufficient dilated linear matrix inequality (LMI) conditions for the \(H_\infty\) static output feedback control problem of linear continuous-time systems with no uncertainty are proposed. The used technique easily and successfully extends to systems with polytopic uncertainties, by means of parameter-dependent Lyapunov functions (PDLFs). In order to reduce the conservatism existing in early standard LMI methods, auxiliary slack variables with even more relaxed structure are employed. It is shown that these slack variables provide additional flexibility to the solution. It is also shown, in this paper, that the proposed dilated LMI-based conditions always encompass the standard LMI-based ones. Numerical examples are given to illustrate the merits of the proposed method.

Cited in 1 Document

MSC:

93B36 \(H^\infty\)-control
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
PDF BibTeX XML Cite
\textit{K. Dabboussi} and \textit{J. Zrida}, J. Appl. Math. 2012, Article ID 812920, 13 p. (2012; Zbl 1235.93085)
Full Text: DOI

References:

[1] V. Blondel and J. N. Tsitsiklis, “NP-hardness of some linear control design problems,” SIAM Journal on Control and Optimization, vol. 35, no. 6, pp. 2118-2127, 1997. · Zbl 0892.93050
[2] V. Ku\vcera and C. E. de Souza, “A necessary and sufficient condition for output feedback stabilizability,” Automatica, vol. 31, no. 9, pp. 1357-1359, 1995. · Zbl 0831.93056
[3] D. Rosinová, V. Veselý, and V. Ku\vcera, “A necessary and sufficient condition for static output feedback stabilizability of linear discrete-time systems,” Kybernetika, vol. 39, no. 4, pp. 447-459, 2003. · Zbl 1249.93150
[4] A. Trofino-Neto and V. Ku\vcera, “Stabilization via static output feedback,” IEEE Transactions on Automatic Control, vol. 38, no. 5, pp. 764-765, 1993. · Zbl 0785.93078
[5] L. El Ghaoui and P. Gahinet, “Rank minimization under LMI constraints: a framework for output feedback problems,” in Proceedings of the European Control Conference (ECC ’93), 1993. · Zbl 0887.93017
[6] D. Henrion and V. Kucera, “Polynomial matrices, LMIs and static output feedback,” in Proceedings of the IFAC/IEEE Symposium on System Structure and Control, Prague, Czech Republic, August 2001. · Zbl 0857.93078
[7] P. Apkarian and D. Noll, “Nonsmooth H\infty synthesis,” IEEE Transactions on Automatic Control, vol. 51, no. 1, pp. 71-86, 2006. · Zbl 1120.93315
[8] Y.-Y. Cao, J. Lam, and Y.-X. Sun, “Static output feedback stabilization: an ILMI approach,” Automatica, vol. 34, no. 12, pp. 1641-1645, 1998. · Zbl 0937.93039
[9] L. El Ghaoui, F. Oustry, and M. AitRami, “A cone complementarity linearization algorithm for static output-feedback and related problems,” IEEE Transactions on Automatic Control, vol. 42, no. 8, pp. 1171-1176, 1997. · Zbl 0887.93017
[10] F. Leibfritz, “A LMI-based algorithm for designing suboptimal static H2/H\infty output feedback controllers,” SIAM Journal on Control and Optimization, vol. 39, no. 6, pp. 1711-1735, 2001. · Zbl 0997.93032
[11] L. Zhang and E.-K. Boukas, “H\infty control for discrete-time Markovian jump linear systems with partly unknown transition probabilities,” International Journal of Robust and Nonlinear Control, vol. 19, no. 5, pp. 868-883, 2009. · Zbl 1166.93320
[12] L. Zhang, P. Shi, and E.-K. Boukas, “H\infty output-feedback control for switched linear discrete-time systems with time-varying delays,” International Journal of Control, vol. 80, no. 8, pp. 1354-1365, 2007. · Zbl 1133.93316
[13] G. I. Bara and M. Boutayeb, “Static output feedback stabilization with H\infty performance for linear discrete-time systems,” IEEE Transactions on Automatic Control, vol. 50, no. 2, pp. 250-254, 2005. · Zbl 1365.93434
[14] C. A. R. Crusius and A. Trofino, “Sufficient LMI conditions for output feedback control problems,” IEEE Transactions on Automatic Control, vol. 44, no. 5, pp. 1053-1057, 1999. · Zbl 0956.93028
[15] Y. Ebihara, D. Peaucelle, and D. Arzelier, “Some conditions for convexifying static H\infty control problems,” Tech. Rep. LAAS-CNRS no 10684, IFAC World Congress, Milano, Italy, 2011. · Zbl 1209.93092
[16] J. C. Geromel, P. L. D. Peres, and S. R. Souza, “Convex analysis of output feedback control problems: robust stability and performance,” IEEE Transactions on Automatic Control, vol. 41, no. 7, pp. 997-1003, 1996. · Zbl 0857.93078
[17] M. Mattei, “Sufficient conditions for the synthesis of H\infty fixed-order controllers,” International Journal of Robust and Nonlinear Control, vol. 10, no. 15, pp. 1237-1248, 2000. · Zbl 0973.93007
[18] E. Prempain and I. Postlethwaite, “Static H\infty loop shaping control of a fly-by-wire helicopter,” in Proceedings of the IEEE Conference on Decision and Control, December 2004. · Zbl 1093.93512
[19] C. E. de Souza and A. Trofino, “An LMI approach to stabilization of linear discrete-time periodic systems,” International Journal of Control, vol. 73, no. 8, pp. 696-703, 2000. · Zbl 1006.93578
[20] P. Apkarian, H. D. Tuan, and J. Bernussou, “Continuous-time analysis, eigenstructure assignment, and H2 synthesis with enhanced linear matrix inequalities (LMI) characterizations,” IEEE Transactions on Automatic Control, vol. 46, no. 12, pp. 1941-1946, 2001. · Zbl 1003.93016
[21] J. Daafouz and J. Bernussou, “Parameter dependent Lyapunov functions for discrete time systems with time varying parametric uncertainties,” Systems & Control Letters, vol. 43, no. 5, pp. 355-359, 2001. · Zbl 0978.93070
[22] K. Dabboussi and J. Zrida, “H2 static output feedback stabilization of linear continuous-time systems with a dilated LMI approach,” in Proceedings of the 6th International Multi-Conference on Systems, Signals and Devices (SSD ’09), March 2009. · Zbl 0978.93070
[23] Y. Ebihara and T. Hagiwara, “New dilated LMI characterizations for continuous-time control design and robust multiobjective control,” in Proceedings of the American Control Conference (ACC ’03), 2003. · Zbl 1061.93040
[24] Y. Ebihara and T. Hagiwara, “New dilated LMI characterizations for continuous-time multiobjective controller synthesis,” Automatica, vol. 40, no. 11, pp. 2003-2009, 2004. · Zbl 1061.93040
[25] K. H. Lee, J. H. Lee, and W. H. Kwon, “Sufficient LMI conditions for H\infty output feedback stabilization of linear discrete-time systems,” IEEE Transactions on Automatic Control, vol. 51, no. 4, pp. 675-680, 2006. · Zbl 1366.93505
[26] M. C. de Oliveira, J. Bernussou, and J. C. Geromel, “A new discrete-time robust stability condition,” Systems & Control Letters, vol. 37, no. 4, pp. 261-265, 1999. · Zbl 0948.93058
[27] M. C. de Oliveira, J. C. Geromel, and J. Bernussou, “Extended H2 and H\infty norm characterizations and controller parametrizations for discrete-time systems,” International Journal of Control, vol. 75, no. 9, pp. 666-679, 2002. · Zbl 1029.93020
[28] D. Peaucelle, D. Arzelier, O. Bachelier, and J. Bernussou, “A new robust D-stability condition for real convex polytopic uncertainty,” Systems & Control Letters, vol. 40, no. 1, pp. 21-30, 2000. · Zbl 0977.93067
[29] U. Shaked, “Improved LMI representations for the analysis and the design of continuous-time systems with polytopic type uncertainty,” IEEE Transactions on Automatic Control, vol. 46, no. 4, pp. 652-656, 2001. · Zbl 1031.93130
[30] J. Zrida and K. Dabboussi, “New dilated LMI characterization for the multiobjective full-order dynamic output feedback synthesis problem,” Journal of Inequalities and Applications, vol. 2010, Article ID 608374, 21 pages, 2010. · Zbl 1204.93055
[31] J. Dong and G.-H. Yang, “Static output feedback control synthesis for linear systems with time-invariant parametric uncertainties,” IEEE Transactions on Automatic Control, vol. 52, no. 10, pp. 1930-1936, 2007. · Zbl 1366.93460
[32] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, vol. 15 of SIAM Studies in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA, 1994. · Zbl 0816.93004
[33] P. Gahinet and P. Apkarian, “A linear matrix inequality approach to H\infty control,” International Journal of Robust and Nonlinear Control, vol. 4, no. 4, pp. 421-448, 1994. · Zbl 0808.93024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.