Robust LMIs with parameters in multi-simplex: Existence of solutions and applications

This paper presents new results concerning the existence of solutions for robust (parameter-dependent) LMIs with parameters lying in a Cartesian product of simplexes, called multi-simplex. These results allow to derive convergent procedures based on LMI relaxations to check the positivity of polynomial matrices with parameters in multi-simplexes. As an application, the robust stability analysis of uncertain linear systems is investigated. As an immediate advantage of this flexible representation, polynomially parameter-dependent Lyapunov functions can be constructed to handle simultaneously time-invariant, arbitrarily time-varying and bounded time-varying parameters in an appropriate way. Numerical experiments illustrate the advantages of the method.