LQ feedback formulation for discrete-time H∞ output feedback

It is well-known that using the standard completion of the squares (CoS) formula for the LQ regulator problem, the H_∞ controller derivations and associated H_∞ controller existence conditions can be directly obtained. In standard theory for general H_∞ output-feedback control and in the associated software, certain preliminary "loop-shifting" transformations are needed which it turns out add unnecessarily to the complexity of the H_∞ theory and to the problem of implementing the H_∞ control theory in software. In this paper, we revisit the CoS derivations for the general H_∞ output feedback problem without the usual assumptions or "loop-shifting" transformations. Similar to earlier work on continuous time case, our results demonstrate that with suitable notation the general case can be handled directly without loop shifting in discrete-time as well. In-fact, we notice that the formulae for the discrete-time case bear semblance to the continuous time formulae with appropriate modifications. Thus we are able to simplifying both the representation and the computation of general discrete-time H_∞ output-feedback controllers.