Output Regulation of Second-Order Nonlinear Systems Subject to Unknown Control Direction Using Extremum Seeking Control

The Nussbaum gain approach has been the standard technique in solving unknown control direction problems. In this paper, we propose control laws composed of extremum seeking control, an internal model, and a compensator signal to solve the robust practical output regulation problem of a second-order system subject to an unknown control direction. Using the Lie bracket approximation technique, we show that the closed-loop system is bounded and the origin is E-Semi-global Practical Uniform Asymptotic Stable. Finally, we illustrate the effectiveness of the proposed approach with a numerical example of a Van der Pol system.