Parametrized LMIs in control theory

A wide variety of problems in control system theory fall within the class of parametrized linear matrix inequalities (LMIs), that is, LMIs whose coefficients are functions of a parameter confined to a compact set. Such problems, though convex, involve an infinite set of LMI constraints, hence are inherently difficult to solve numerically. This paper investigates relaxations of parametrized LMI problems into standard LMI problems using techniques relying on directional convexity concepts. An in-depth discussion of the impacts of the proposed techniques in quadratic programming, Lyapunov-based stability and performance analysis, μ analysis and linear parameter varying control is provided. Illustrative examples are given to demonstrate the usefulness and practicality of the approach