Optimization of the loop transfer function†

Abstract

In quantitative feedback synthesis, the objective is to satisfy assigned performance tolerances over given ranges of plant uncertainty and external disturbances. In such linear and non-linear problems, whether single, multiple-loop or multivariable, the synthesis techniques result in frequency-domain bounds ψ_i(ω) in the complex plane, on the loop transmission functions L_i(jω). This paper presents a simple proof that an optimum L_i(jω) lies on its ψ_i(ω) for each ω∊[0, ∞). Also, a numerical technique is presented for deriving any desired approximation to the optimum.