Adaptive control for a discrete-time first-order nonlinear system with both parametric and non-parametric uncertainties

A simple first-order discrete-time nonlinear system, which has both parametric uncertainty and non parametric uncertainty, is studied in this paper. The uncertainty of non-parametric part is characterized by a Lipschitz constant L, and the nonlinearity of parametric part is characterized by an exponent index b. An adaptive controller is constructed for this model in both cases of b = 1 and b > 1, and its closed-loop stability is established under some conditions. When b = 1, the conditions given reveal the magic number 3/2 + √2 which appeared in previous study on capability and limitations of the feedback mechanism.