Feedback systems with non-linear uncertain plants†

Two non-linear feedback problems are considered. In one, a non-linear plant of net order e, with highly uncertain parameters, is subjected to external disturbances of bounded magnitude, and of bounded variation over any finite interval [0, T]. It is shown how causal LTI (linear time invariant) compensation can guarantee the resulting output y(t), and its derivatives y(u)(t) for u≤e-1 can be made arbitrarily small in magnitude. The applicable non-linear class is greatly increased by use of non-linear compensation. Owing to the finite [0, T] interval, there is no theoretical difference between minimum and non-minimum-phase plants, both for LTI and nonlinear systems. In the second problem there is an infinite set of command inputs 𝒸= {c(trpar; }. It is required that the closed-loop system, with its uncertain non-linear plant, has an output ylpar;trpar;, which can be written as y=φ*c, a linear convolution, for any cϵ𝒸, with φϵΨa set of acceptable response functions. The tolerances on Φcan be arbitrarily narro...