Robust H_∞ performance using lifted polynomial parameter-dependent Lyapunov functions

Abstract

This paper investigates the robust H _∞ performance of time-invariant linear uncertain systems where the uncertainty is in polytopic domains. Robust H _∞ is checked by constructing a quadratic parameter-dependent Lyapunov function. The matrix associated with this quadratic Lyapunov function is a polynomial function of the uncertain parameters, expressed as a particular polynomial matrix involving κ powers of the dynamic matrix of the system and one symmetric matrix to be determined. The degree of this polynomial matrix function is arbitrary. Finsler's Lemma is used to lift the obtained stability conditions into a larger space in which sufficient stability tests can be developed in the form of linear matrix inequalities. As κ increases, less conservative H _∞ evaluations are obtained. Both continuous and discrete-time systems are investigated. Numerical examples illustrate the method and compare the present results with similar works in the literature.

Acknowledgments

This work has been partially supported by the Brazilian agencies CNPq and FAPESP. The authors wish to thank the anonymous reviewers for suggestions that helped to improve the paper.