State-space solutions to standard H2 and H ∞ control problems

Simple state-space formulas are derived for all controllers solving the following standard H_∞ problem: For a given number γ>0, find all controllers such that the H _∞ norm of the closed-loop transfer function is (strictly) less than γ. It is known that a controller exists if and only if the unique stabilizing solutions to two algebraic Riccati equations are positive definite and the spectral radius of their product is less than γ². Under these conditions, a parameterization of all controllers solving the problem is given as a linear fractional transformation (LFT) on a contractive, stable, free parameter. The state dimension of the coefficient matrix for the LFT, constructed using the two Riccati solutions, equals that of the plant and has a separation structure reminiscent of classical LQG (i.e. H ₂) theory. This paper is intended to be of tutorial value, so a standard H₂ solution is developed in parallel