Feedback systems with non-linear uncertain plants†

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Abstract
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 narrow. This can be acheived by LTI compensation, for one class of non-linear uncertain plants. Again, the applicable class is greatly increased by use of non-linear compensation. In both cases, the important result- is that just as in LTI systems, it is simply a matter of having a loop transmission function which is sufficiently large over a large enough bandwidth and with suitable stability margins. The relevant loop transmission is, in fact, LTI in this paper, despite the highly non-linear and uncertain plant.

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