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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H∞</tex> 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H∞</tex> 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</tex> or "approximate-inverse" parametrization of feedbacks that maintain closed-loop stability is extended to the ease of unstable plants. The <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H∞</tex> and Wiener-Hopf approaches are compared.