Optimization of the loop transfer function†

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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 ψi(ω) in the complex plane, on the loop transmission functions Li(jω). This paper presents a simple proof that an optimum Li(jω) lies on its ψi(ω) for each ω∊[0, ∞). Also, a numerical technique is presented for deriving any desired approximation to the optimum.