In this paper, we study stability and performance properties of linear fractional transformation (LFT) parameter-dependent systems using duality theory and tools from convex analysis. A pair of conjugate functions, the convex hull and the maximum of a family of quadratic functions, are used for analysis and synthesis of LFT systems. Sufficient synthesis conditions for both robust state feedback and gain-scheduling output feedback control problems are formulated as a set of linear matrix inequalities (LMIs) with linear search over scalar variables. Finally, a numerical example is used to demonstrate the advantages of the proposed approaches.