Quantitative feedback theory

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Abstract
In quantitative feedback theory, plant parameter and disturbance uncertainty are the reasons for using feedback. They are defined by means of a set Q = {P} of plant operators and a set D = {D} of disturbances. The desired system performance is defined by sets of acceptable outputs Au in response to an input u, to be achieved for all P∈ Q. If any design freedom remains in the achievement of the design specifications, it is used to minimise the effect of sensor noise at the plant input. Rigorous, exact quantitative synthesis theories have been established to a fair extent for highly uncertain linear, nonlinear and time-varying single-input single-output, single-loop and some multiple-loop structures; also for multiple-input multiple-output plants with output feedback and with internal variable feedback, both linear and nonlinear. There have been many design examples vindicating the theory. Frequency-response methods have been found to be especially useful and transparent, enabling the designer to see the trade-off between conflicting design factors. The key tool in dealing with uncertain nonlinear and multiple-input multiple-output plants is their conversion into equivalent uncertain linear time-invariant single-input single-output plants. Schauder's fixed-point theorem justifies the equivalence. Modern control theory, in particular singular-value theory, is examined and judged to be comparatively inadequate for dealing with plant parameter uncertainties.

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