The H2 Control Problem: State-space and Transfer-function Solutions

The H2 control problem consists of stabilizing the control system while minimizing the H2 norm of its transfer function. Several solutions to this problem are available. For systems in state space form, an optimal regulator can be obtained in observer form by solving two algebraic Riccati equations. For systems described by transfer functions, either Wiener-Hopf optimization or projection results can be applied. The optimal regulator is then obtained using operations with proper stable rational matrices: inner-outer factorizations and stable projections. The aim of this paper is to compare the two approaches. It is well understood that the inner-outer factorization is equivalent to solving an algebraic Riccati equation. However, why are the stable projections not needed in the state-space approach? The difference between the two approaches derives from a different construction of doubly coprime proper stable factors used to represent the plant. The transfer-function approach takes any fixed doubly coprime factors, while the state-space approach parametrizes all such factors and those selected obviate the need for stable projections.