Adaptive nonlinear regulation for uncertain minimum phase systems with unknown exosystem

The theory of regulation aims at zero output regulation error under the assumption that the order of the exosystem, which generates the disturbance and the output reference signals, or at least its upper bound is known; this is a crucial assumption since the regulator which achieves exact regulation should incorporate a possibly adaptive internal model which includes the exosystem. We design an adaptive internal model to characterize the output regulation error obtained on the basis of the features of the ideal unknown input reference signal capable of exactly zeroing the regulation error. We show that if the designed adaptive internal model can generate the ideal zeroing reference input, then the regulation error tends to zero while if the adaptive internal model cannot generate the ideal zeroing reference input, then the regulation error exponentially tends to a residual which decreases with the number of unmodeled terms in the ideal reference input.