Abstract The set of H ∞ controllers with closed-loop performance γ can be implicitly parametrized by the solutions ( R, S) of a system of linear matrix inequalities (LMI). The matrices R and S play a role analogous to that of the Riccati solutions X ∞ and Y ∞ in classical Riccati-based H ∞ control. Useful applications include LMI-based H ∞ synthesis, mixed H 2 H ∞ design, and H ∞ design with a pole-placement constraint. This paper is concerned with the reliable computation of H ∞ controllers given a solution ( R, S) of the characteristic system of LMIs. Explicit formulas are derived for both the regular and singular cases. Remarkably, these formulas are extensions of the usual ‘central controller’ formulas where the LMI solutions R and S replace the Riccati solutions X ∞ and Y ∞. Simple and numerically appealing new formulas for discrete-time H ∞ controllers are also derived.