Non-linear uncertain feedback systems with initial state values†

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Abstract
A single input-output non-linear plant w is considered which, due to uncertainty, is known only to be a member of a set . At t = 0, the plant has initial state values given by a vector Y 0, member of a defined set 0. At this instant it is switched into a feedback structure with compensation function G, which has zero initial state values. For certain classes of  it is shown how G may be designed to achieve arbitrary fast attenuation of the plant output for all wε Y 0ε0. The design philosophy is to convert this problem into one of disturbance attenuation in a linear time-invariant plant P with zero initial states ; Pε𝒫 a set of linear time-invariant plants and the disturbance dε a set of disturbances. ,𝒫 are determined by 0,  and the performance specifications. Schauder's fixed point theorem is used to rigorously justify this technique. The design procedure may be readily used by engineers who are quite ignorant of the mathematical theory of non-linear operators. Ordinary, familiar frequency-response concepts are used in design execution, A design example is included.