Robust stability and performance of linear time-varying systems in polytopic domains

Abstract

A sufficient condition to assess the stability of linear time-varying systems in polytopic domains is given in this paper. This condition is written as a feasibility test of linear matrix inequalities expressed at the polytope vertices and taking bounds on the time-derivatives of the system parameters into account. Differently from other techniques in the literature, there is no need of gridding procedures on the parameter space neither restrictive assumptions on the uncertainty structure. The proposed test can be solved in polynomial time yielding as solution a parameter dependent Lyapunov function that assures the system stability. Moreover, the results can be applied to systems with affine parameter uncertainty and can be easily extended to deal with

guaranteed cost computation. Numerical examples show that the proposed condition can lead to less conservative evaluations of the stability domain and of the performance index for this class of systems when compared to the results based on the conventional quadratic stability or to the ones from similar techniques in the literature.

Acknowledgements

This work has been partially supported by the Brazilian agencies CAPES, CNPq and FAPESP. The authors would like to thank the reviewers for their valuable suggestions.

Notes

The LMIs must be implemented with all signal combinations of ±, yielding N2^{( N   − 1)} LMIs in (6) and N(N − 1)2^{( N   − 2)} LMIs in (7).

The time dependency of α(t) will be omitted in the proof, for sake of simplicity.