Loop transfer recovery for general observer architectures

A general and concise formulation is given of the loop transfer recovery (LTR) design problem based on recovery errors. Three types of recovery errors are treated: open loop recovery, sensitivity recovery and input-output recovery errors. The three corresponding versions of the asymptotic recovery problem tum out to be equivalent, since the minimization of the recovery errors all amount to the minimization of a certain matrix, the recovery matrix. Using the recovery error definitions, simple necessary and sufficient conditions for the controllers are derived both for the exact and asymptotic recovery cases. This general recovery formulation covers all known observer based compensator types as special cases. The conditions given in this setting are effectively the aim of all known LTR design methods. The recovery formulation is interpreted in terms of a model-matching problem as well, which is examined by means of the Q-parametrization. It is shown how the general controller obtained by the Q-parametrization can be written as a Luenberger observer based controller. In all cases, n controller states suffice to achieve recovery. The compensators are characterized for errors both on the input- and on the output-node (dual case). I