On direct adaptive control of a class of nonlinear scalar systems

We develop an adaptive control for a scalar system with nonlinear uncertainty. Specifically, we consider scalar systems that are not stabilizable via a linear feedback and hence there does not exist a simple procedure to design stabilizing adaptive controllers for these systems. We present an adaptive control framework for such scalar systems that guarantees convergence of the state to the origin. The overall objective of this note is to emphasize the inherent difficulties in designing adaptive controllers for systems with nonlinear uncertainties.