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

Abstract

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. H2) theory. This paper is intended to be of tutorial value, so a standard H2 solution is developed in parallel.