This paper proposes a method for improving the LMI-based robust control synthesis. It is well-known that the design of a H∞ controller can be formulated as a two-step procedure. In the first step the performance level is minimized and the closed loop Lyapunov function is determined. The second step is the construction of the controller. The currently available implementations of this procedure generate only one controller candidate and leave no room for additional tuning between the design steps. If the controller obtained is not acceptable, (e.g. it is unstable) then the designer has to modify the original control problem (changing, e.g., the performance weighting functions) and then has to perform the entire synthesis again. This paper presents a novel parameterization of all controllers corresponding to the performance level and Lyapunov function determined in the first design step. This parameterizaton makes it possible to construct tuning algorithms for finding the 'best' controller in the parameterized set. Now, this paper focuses only on the stability of the controller and provides an iterative tuning algorithm, which finds a stable controller having poles on the prescribed domain of the left half complex plane. The applicability of the proposed algorithm is demonstrated on a simple robust control problem.