Output feedback adaptive stabilization of second-order systems

We consider output feedback adaptive stabilization for second-order systems with no zeros. The assumptions we make are standard, namely, that the sign of the high frequency gain is known. However, we complement the existing literature by deriving an explicit expression for the adaptive controller. The controller has the form of a 6th-order dynamic compensator with quadratic, cubic and quartic nonlinearities. The proof of convergence is based on a variation of Lyapunov´s method in which the Lyapunov derivative is shown to be asymptotically nonpositive. Application of the controller to the Van der Pol and Duffing oscillators shows that the controller is effective for nonlinear systems as well