A separation theorem for guaranteed H₂ performance through matrix inequalities

The usage of convex optimisation programs that leverage linear matrix inequalities allows for numerical solutions to the design of output-feedback controllers with guaranteed H₂ performance. As decreed by the classical separation theorem for the related LQG control problem, the H₂ control problem admits an optimal solution in terms of those of the separate optimal state-estimation and state-feedback design problems. This work details a new and alternative proof of this separation theorem. The proof builds on techniques for (linear) matrix inequalities and shows, in particular, that feasible but sub-optimal solutions of the state-feedback and the state-estimation problem yield a sub-optimal output feedback controller with guaranteed H₂ performance.