Model-based extremum-seeking control for unstable systems with time-varying extremum

This paper addresses the adaptive extremum-seeking problem for a class of nonlinear systems with time-varying extremum. The proposed technique can be used to identify the optimum trajectory for a class of unstable nonlinear dynamics with an uncertain objective function. A geometric-based adaptive scheme is developed for estimate the unknown parameters of the dynamical system. A gradient-based approach is designed to solve the uncertain optimization problem. This method guarantees uniform boundedness of the closed-loop signals and it is shown that the closed-loop trajectories of the system can track an unknown time-varying optimum. The effectiveness of the proposed method is illustrated with a simulation example.