Explicit controller formulas for LMI-based H/sub /spl infin// synthesis

The set of H/sub /spl infin// controllers with closed-loop performance /spl gamma/ 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/sub /spl infin// and Y/sub /spl infin// in classical state-space H/sub /spl infin// control. Useful applications of this parametrization include reduced-order H/sub /spl infin// synthesis, mixed H/sub 2//H/sub /spl infin// design, H/sub /spl infin// design with a pole placement constraint, etc. This paper is concerned with the computation of H/sub /spl infin// 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/sub /spl infin//, Y/sub /spl infin//. In addition, they apply to regular as well as singular H/sub /spl infin// problems. Finally, the LMI-based approach also leads to simple and numerically appealing new formulas for discrete-time H/sub /spl infin// controllers.