Feedback, minimax sensitivity, and optimal robustness

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 to disturbances and robustness under plant perturbations are 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 case is undertaken. The results are extended to unstable plants, and explicit formulas for the general situation of a finite number of right half-plane (RHP) plant zeros or poles are provided. The Q or "approximate-inverse" parametrization of feedbacks that maintain closed-loop stability is extended to the ease of unstable plants. The H^ and Wiener-Hopf approaches are compared.