Positive polynomial matrices and improved LMI robustness conditions

Recently several new LMI conditions for stability of linear systems have been proposed, introducing additional slack variables to reduce the gap between conservative convex quadratic stability conditions and intractable non-convex robust stability conditions. In this paper we show that these improved LMI conditions can be derived with the help of some basic results on positive polynomial matrices, providing a clear interpretation of the role of the additional variables. The approach allows to derive in a unifying way results in the state-space and polynomial frameworks. Applications to robust stability analysis and robust stabilization of systems with multi-linear parametric uncertainty are fully described.