On General H2 Control: From Frequency To Time Domain

This article presents a general formula for discrete-time H2 control. It works with the regular and singular case of H2 control, i.e. in the case of possibly non-left-invertible matrices G12 and non-right-invertible matrices G21, with possible unit circle invariant zeros. In the generic case, it can be simplified and adapted to work with the plant transfer matrix directly, without invoking the matrices of the parameterisation of stabilising controllers. A further result of this article is the presented necessary and sufficient conditions for state-space H2 control, under only stabilisability and detectability assumptions. If the conditions are satisfied, an observer-based H2 controller is constructed. The corresponding numerical algorithm consists of solving two discrete-time algebraic Riccati systems (DARSs) and two eigen-problems.