The problem of designing a controller to meet di erent speci cations or deciding that no such controller exists is addressed in this paper by linking three types of tools: the Youla pa- rameterization allows searching for a controller in a convex set; formulations using Linear Matrix Inequalities (LMI) are proposed for di erent practical speci cations; the correspond- ing convex problem is solved using a Cutting Plane Algorithm (CPA). Such an approach is developed by overcoming the problem of the huge number of additional variables which often occurs in the LMI framework, particulary when used in conjunction with the Youla parameterization. Its e ciency is discussed by considering two practical control problems.