Comparison between H2 and H∞ controllers

The H_∞ performances of a controller designed by H ₂ methodology and a controller designed by H_∞ methodology are quantitatively compared. It is shown that the ratio of the H_∞ performance of the two is no greater than 2k for the one block problem, where k is the total number of right half plane transmission zeros for two subsystems namely the subsystem between the disturbance and the measurement and the subsystem between the control signal and the controlled output. It is also shown that there are systems which can give a ratio arbitrarily close to the upper bound. However, those systems are of a pathological type. Some possible generalizations to two- and four-block problems are discussed