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A new robust D-stability condition for real convex polytopic uncertainty. (English) Zbl 0977.93067
Summary: The problem of robust D-stability analysis with respect to real convex polytopic uncertainties is tackled. A new LMI-based sufficient condition for the existence of parameter-dependent Lyapunov functions is proposed. This condition generalises previously published conditions. Numerical comparisons with quadratic stability results as well as previous results based on parameter-dependent Lyapunov functions illustrate the relevance of this new condition. Finally, this result appears to be promising for robust multi-objective performance analysis and control synthesis purposes.

##### MSC:
 93D09 Robust stability 93D30 Lyapunov and storage functions 15A39 Linear inequalities of matrices
##### Software:
LMI toolbox; RoMulOC; Sdpsol
Full Text:
##### References:
 [1] D.Arzelier, D. Peaucelle, Robust multi-objective output-feedback control for real parametric uncertainties via parameter-dependent Lyapunov functions, ROCOND 2000, September 1999, submitted for publication. · Zbl 1051.93078 [2] Bachelier, O.; Pradin, B., Bounds for uncertain matrix root-clustering in a union of subregions, Int. J. robust non-linear control, 9, 6, 333-359, (1999) · Zbl 0931.93019 [3] Barmish, B.R.; Kang, H.I., A survey of extreme point results for robustness of control systems, Automatica, 29, 13-35, (1993) · Zbl 0772.93022 [4] Boyd, S.; El Ghaoui, L.; Feron, E.; Balakrishnan, V., Linear matrix inequalities in system and control theory, (1994), SIAM Studies in Applied Mathematics Philadelphia, PA · Zbl 0816.93004 [5] Chilali, M.; Gahinet, P., $$H∞$$ design with pole placement constraints: an $$LMI$$ approach, IEEE trans. automat. control, AC-41, 3, 358-367, (1996) · Zbl 0857.93048 [6] Feron, E.; Apkarian, P.; Gahinet, P., Analysis and synthesis of robust control systems via parameter-dependent Lyapunov functions, IEEE trans. automat. control, AC-41, 7, 1041-1046, (1996) · Zbl 0857.93088 [7] P. Gahinet, A. Nemirovski, A.J. Laub, M. Chilali, $$LMI$$ Control Toolbox User’s Guide, The Mathworks Partner Series, Natick, MA, 1995. [8] Gahinet, P.; Apkarian, P.; Chilali, M., Affine parameter-dependent Lyapunov functions and real parametric uncertainty, IEEE trans. automat. control, AC-41, 3, 436-442, (1996) · Zbl 0854.93113 [9] Geromel, J.C.; Peres, P.L.D.; Bernussou, J., On a convex parameter space method for linear control design of uncertain systems, SIAM J. control opt., 29, 381-402, (1991) · Zbl 0741.93020 [10] Geromel, J.C.; de Oliveira, M.C.; Hsu, L., LMI characterization of structural and robust stability, Linear algebra appl., 285, 1-3, 69-80, (1998) · Zbl 0949.93064 [11] Gutman, S.; Jury, E.I., A general theory for matrix root clustering in subregion of the complex plane, IEEE trans. automat. control, AC-26, 853-863, (1981) · Zbl 1069.93518 [12] Iwasaki, T.; Hara, S., Well-posedness of feedback systems: insights into exact robustness analysis and approximate computations, IEEE trans. automat. control, 43, 5, 619-630, (1998) · Zbl 0927.93038 [13] A.G. Mazco, The Lyapunov matrix equation for a certain class of regions bounded by algebraic curves, Sov. Automat. Control (13) (1980) 37-42. [14] de Oliveira, M.C.; Bernussou, J.; Geromel, J.C., A new discrete-time robust stability condition, Systems control lett., 37, 4, 261-265, (1999) · Zbl 0948.93058 [15] de Oliveira, M.C.; Geromel, J.C.; Hsu, L., LMI characterization of structural and robust stability: the discrete-time case, Linear algebra appl., 296, 1-3, 27-38, (1999) · Zbl 0949.93063 [16] D. Peaucelle, D. Arzelier, D. Henrion, Performance and quadratic stabilisability via dynamic output feedback for uncertain generalised models, Proceedings of the 14th World IFAC Congress, Beijing, P.R. China, July 1999. [17] D. Peaucelle, D. Arzelier, Robust performance analysis $$LMI$$-based methods for real parametric uncertainty via parameter-dependent Lyapunov functions, IEEE Trans. Automat. Control, submitted. · Zbl 1051.93078 [18] Scherer, C.; Gahinet, P.; Chilali, M., Multiobjective output-feedback control via $$LMI$$ optimisation, IEEE trans. automat. control, 42, 7, 896-911, (1997) · Zbl 0883.93024 [19] S.P. Wu, S. Boyd, Design and implementation of a parser/solver for SDP with matrix structure, IEEE Conference on Computer Aided Control System Design, 1996. [20] Zhou, K.; Doyle, J.C.; Glover, K., Robust and optimal control, (1996), Prentice-Hall Englewood Cliffs, NJ · Zbl 0999.49500
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