Generalized uncertainty and quadratic stabilizability: an LMI approach

A considerable amount of work has been devoted to the problem of stabilizing uncertain dynamic systems with bounded uncertainties. These works are based on the most fundamental results concerning stability of feedback systems which are the small gain theorem and the positivity theorem. Each of these results was established using a Lyapunov function framework. Specifically, necessary and sufficient conditions of quadratic stability for positive real or sector-bounded feedback were derived. In this article, an alternative model for real parameter uncertainty is proposed allowing the unification of the existing different points of view in a single generalized framework. Here, the modeling of structured uncertainty is defined via a matrix inequality relating the input and the output of the perturbations acting on the nominal system. All the results of this article are based upon the quadratic stabilizability concept. Necessary and sufficient conditions of quadratic stability and quadratic stabilizability are then derived in terms of the existence of solutions to a linear matrix inequality. A new setup for LTI systems which generalizes passivity setup or β-bounded setup is also defined. In such a framework, a transfer function is supposed to verify a generalized sector constraint associated with a symmetric matrix. A state-space characterization is proposed and equivalence with quadratic stability of the uncertain generalized model is established