A new approach to classical frequency methods: Feedback and minimax sensitivity

In this paper, we look for feedbacks that minimize the sensitivity function of a linear, single-variable feedback system represented by its frequency responses. Sensitivity is measured in a weighted H?? norm. In an earlier paper, Zames proposed an approach to feedback design involving the measurement of sensitivity by "multiplicative seminorms", which have certain advantages over the widely used quadratic norm in problems where there is plant uncertainty, or where signal power-spectra are not fixed but belong to sets. The problem was studied in a general setting, and some H?? examples were solved. Here, a detailed study of the single-variable cases is undertaken. The results are extended to unstable plants, and explicit formulas for the general situation of a finite number of RHP plant zeros or poles are provided. The Q or "approximate-inverse" parametrization of feedbacks that maintain closed-loop stability is extended to the case of unstable plants. The H?? and Wiener-Hopf approaches are compared.