Necessary and sufficient conditions of quadratic stability of uncertain linear systems

The stability of linear systems subject to possibly fast time-varying uncertainties is analyzed. A necessary and sufficient condition of quadratic stability is derived. An uncertainty stability margin coefficient ρ is introduced to give a quantitative measure of the stability. It is proposed that the uncertain region be approximated by a convex hyperpolyhedron. In this case, the computation of ρ becomes a two-level optimization problem, in which the extremum of the inner level can be reached by one of the corners of the hyperpolyhedron