On the generalized H∞ control problem

An essentially non-iterative solution of the so-called Generalized H_∞ (GH_∞) control problem is presented. Closed formulae are given in terms of a real Schur decomposition, the solution of two Lyapunov equation and a single, well-conditioned eigenvalue problem. The approach is based on an embedding of GH_∞ in the Standard H_∞ control problem followed by application of the "two-Riccati equation" H_∞ theory. The results include a characterization of a particular sub-optimal controller and necessary and sufficient conditions for the case where an optimal controller yields a zero norm. One of the limitations of the GH_∞ approach as a practical design tool is illustrated by means of example. It is shown how, for a simple plant (a pure gain), a GH_∞ optimal controller yields a non-robust closed-loop system even for a relatively simple, "sensible" selection of control design weighting functions.