For a given linear time-invariant plant

$P\left(s\right)$ with state-space realization, we have to find an

${H}_{\infty}$-suboptimal, internally stabilizing controller

$u=K\left(s\right)y$ that makes the closed-loop

${L}_{2}$ gain from the disturbance to the error signal less than a prescribed value. The design is based on linear matrix inequalities (LMI). The synthesis is performed in two steps: First, solve a system of three LMIs, where the unknowns are two symmetric matrices

$R$,

$S$ of size equal to the plant order. Second, given

$R$,

$S$ and the plant, compute the controller by solving another LMI called the controller LMI. For the second part the paper finds explicit formulas. These were implemented in the LMI Control Toolbox for use with MATLAB.