A note on a double convex combination matrix inequality

A double convex combination matrix inequality arising in control and filtering problems of dynamic systems is revisited in this paper. Focus is placed on comparisons of conservatism caused by some commonly-used sufficient conditions which ensure the validity of the double convex combination matrix inequality. It is found that the conservatism of analysis and synthesis results of the control and filtering problems for a large class of systems, such as polytopic systems, parameter varying systems as well as Takagi-Sugeno (TS) model based nonlinear systems, mainly involves the sufficient conditions corresponding to the double convex combination matrix inequality. Four sufficient conditions for the validity of the double convex combination matrix inequality are presented. By the matrix-string set approach and analytic tools of negative semi-definiteness of symmetric matrices, detailed comparison results for conservatism of these sufficient conditions are provided.