Robust stability of time varying polytopic systems. (English) Zbl 1129.93479

Summary: This paper provides global asymptotic stability conditions for time-varying continuous-time polytopic systems, using a parameter dependent Lyapunov function. The time varying parameter uncertainty as well as its time derivative are modelled as belonging to polytopic convex sets and their dependence is made explicit to get less conservative results. A particular case, characterized by the parameter uncertainty satisfying a linear differential equation is analyzed and a simpler version of the aforementioned stability conditions is presented. The results are expressed in terms of linear matrix inequalities being thus numerically solvable with no big difficulty. The theory is illustrated by the determination of the asymptotic stability region of a Mathieu’s type equation with an uncertain time varying parameter.


93D09 Robust stability
Full Text: DOI


[1] Almeida, H. L.S.; Bhaya, A.; Falcão, D. M.; Kaszkurewicz, E., A team algorithm for robust stability analysis and control design of uncertain time-varying linear systems using piecewise quadratic Lyapunov functions, Internat. J. Robust Nonlinear Control, 11, 357-371 (2001) · Zbl 0984.93062
[2] Boyd, S. P.; El Ghaoui, L.; Feron, E.; Balakrishnan, V., Linear Matrix Inequalities in System and Control Theory (1994), SIAM: SIAM Philadelphia · Zbl 0816.93004
[3] Colaneri, P.; Geromel, J. C.; Locatelli, A., Control Theory and Design—An \(RH_2\) and \(RH_\infty\) Viewpoint (1997), Academic Press: Academic Press New York
[4] Daafouz, J.; Bernussou, J., Parameter dependent Lyapunov functions for discrete time systems with time varying parametric uncertainties, Systems Control Lett., 43, 355-359 (2001) · Zbl 0978.93070
[5] Gahinet, P.; Apkarian, P.; Chilali, M., Affine parameter dependent Lyapunov functions and real parametric uncertainty, IEEE Trans. Automat. Control, 41, 436-442 (1996) · Zbl 0854.93113
[6] Geromel, J. C.; de Oliveira, M. C.; Hsu, L., LMI characterization of structural and robust stability, Linear Algebra Appl., 285, 69-80 (1998) · Zbl 0949.93064
[7] 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 Optim., 29, 381-402 (1991) · Zbl 0741.93020
[8] Luenberger, D. G., Introduction to Dynamic Systems—Theory, Models and Applications (1979), Wiley: Wiley New York · Zbl 0458.93001
[9] Rockafellar, R., Convex Analysis (1970), Princeton Press: Princeton Press Princeton · Zbl 0193.18401
[10] Trofino, A.; de Souza, C. E., Biquadratic stability of uncertain linear systems, IEEE Trans. Automat. Control, 46, 1303-1307 (2001) · Zbl 1014.93030
[11] Xie, L.; Shishkin, S.; Fu, M., Piecewise Lypaunov functions for robust stability of linear time-varying systems, Systems Control Lett., 31, 165-171 (1997) · Zbl 0901.93063
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.