An elementary state-space approach to the H2 optimal control problem

The existing approaches to the H₂ optimal control problem, such as the one of converting the problem to an H₂ model-matching problem and the one using change of variables and inner functions, are not ideal for pedagogical purposes because they use highly intricate mathematical tools and/or have a complicated proof structure. We here describe a new state-space approach which is elementary and conceptually simple. The key fact underlying our approach is that the linearity of the two dual H₂ norm formulas in form of a Lyapunov equation and a matrix trace allows us to suitably decompose the H₂ norm-square of the closed-loop system into the norm-squares of three subsystems.