Explicit controller formulas for LMI-based H∞ synthesis

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, S play a role analogous to that of the Riccati solutions X_∞ and Y_∞ in classical state-space H_∞ control. Useful applications of this parametrization include reduced-order H_∞ synthesis, mixed H₂/H_∞ design, H_∞ design with a pole placement constraint, etc. This paper is concerned with the computation of H_∞ controllers given any solution (R, S) of the characteristic system of LMIs. Explicit and numerically reliable formulas are derived for both full- and reduced-order cases. Remarkably, these formulas turn out to be simple extensions of the usual "central controller" formulas where the LMI solutions R, S replace the Riccati solutions X_∞, Y_∞. In addition, they apply to regular as well as singular H_∞ problems. Finally, the LMI-based approach also leads to simple and numerically appealing new formulas for discrete-time H_∞ controllers.