An LMI condition for robust stability of polynomial matrix polytopes. (English) Zbl 0982.93057

Henrion, Didier; Arzelier, Denis; Peaucelle, Dimitry; Šebek, Michael

Automatica 37, No. 3, 461-468 (2001).

The authors propose a new sufficient condition for robust stability of polynomial matrix polytopes. They combine two results obtained previously for less complex problems, namely a stability test for polynomial matrices expressed in the form of the LMI feasibility problem and a robust stability condition for constant matrix polytopes [see J. C. Geromel, M. C. de Oliveira and L. Hsu, LMI characterization of structural and robust stability, Linear Algebra Appl. 285, 69-80 (1998; Zbl 0949.93064)] based on the idea of an extra degree of freedom introduced by an additional arbitrary matrix enabling linear dependence of the condition in the parameters. This in turn leads to reduction of conservatism inherent to the quadratic stability conditions. The stability regions considered in the paper include the left half plane and the unit disk so that the obtained condition may be used to check robust stability of both continuous-time and discrete-time multivariable systems. The authors also present the results of their numerical experiments in which they used the LMI control toolbox of MATLAB. The results prove the reduction of the conservatism inherent to the quadratic stability tests.

Reviewer: A.Šwierniak (Gliwice)

MSC:

93D09	Robust stability
93C73	Perturbations in control/observation systems
15A39	Linear inequalities of matrices

Keywords:

robust stability; polynomial matrix polytopes; LMI feasibility; quadratic stability

Citations:

Zbl 0949.93064

Software:

Polynomial Toolbox; Control System Toolbox; Matlab; PolyX; LMI toolbox

References:

[1]	Barmish, B. R., Necessary and sufficient conditions for quadratic stabilizability of an uncertain system, Journal of Optimization Theory and Applications, 46, 399-408 (1985) · Zbl 0549.93045
[2]	Barmish, B. R., New tools for robustness of linear systems (1994), Macmillan: Macmillan New York · Zbl 1094.93517
[3]	Bhattacharyya, S. P.; Chapellat, H.; Keel, L. H., Robust control: the parametric approach (1995), Prentice-Hall: Prentice-Hall Upper Saddle River, NJ · Zbl 0838.93008
[4]	Blondel, V. D.; Tsitsiklis, J. N., A survey of computational complexity results in systems and control, Automatica, 36, 9, 1249-1274 (2000) · Zbl 0989.93006
[6]	Chilali, M.; Gahinet, P., \(H∞\) design with pole placement constraints: an LMI approach, IEEE Transactions on Automatic Control, 41, 3, 358-367 (1996) · Zbl 0857.93048
[7]	Geromel, J. C.; Peres, P. L.D.; Bernussou, J., On a convex parameter space method for linear control design of uncertain systems, SIAM Journal on Control and Optimization, 29, 381-402 (1991) · Zbl 0741.93020
[8]	Geromel, J. C.; de Oliveira, M. C.; Hsu, L., LMI characterization of structural and robust stability, Linear Algebra and its Applications, 285, 68-80 (1998) · Zbl 0949.93064
[10]	Kailath, T., Linear systems (1980), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0458.93025
[11]	Karl, W. C.; Verghese, G. C., A sufficient condition for the stability of interval matrix polynomials, IEEE Transactions on Automatic Control, 38, 7, 1139-1143 (1993) · Zbl 0800.93954
[12]	Kučera, V., Discrete linear control: the polynomial equation approach (1979), Wiley: Wiley Chichester, England
[14]	de Oliveira, M. C.; Bernussou, J.; Geromel, J. C., A new discrete-time robust stability condition, Systems and Control Letters, 37, 4, 261-265 (1999) · Zbl 0948.93058
[15]	Peaucelle, D.; Arzelier, D.; Bachelier, O.; Bernussou, J., A new robust \(D\)-stability condition for real convex polytopic uncertainty, Systems and Control Letters, 40, 1, 21-30 (2000) · Zbl 0977.93067