An H∞ Formulation of Quantitative Feedback Theory

The QFT robust performance problem in its entirety may be reduced to an H∞ problem by casting each specification as a frequency domain constraint either on the nominal sensitivity function or the complementary sensitivity function. In order to alleviate the conservative nature of a standard H∞ solution that is obtainable for a plant with parametric uncertainty we develop a new stability criterion to replace the small gain condition. With this new stability criterion it is shown that the existence of a solution to the standard H∞ problem guarantees a solution to the QFT problem. Specifically, we provide an explicit characterization of necessary frequency weighting functions for an H∞ embedding of the QFT specifications. Due to the transparency in selecting the weighting functions, the robust performance constraints can be easily relaxed, if needed, for the purpose of assuring a solution to the H∞ problem. Since this formulation provides only a sufficient condition for the existence of a QFT controller one can then use the resulting H∞ compensator to initiate the QFT loop shaping step.