Constrained Extremum Seeking Controller Design for a Class of Second-Order Nonlinear Systems

This paper investigates the problem of stabilization and output minimization for a class of second-order nonlinear systems with state constraints. We show that integrating a model-based extremum seeking controller (ESC) within the feedback loop guarantees that both the system output and trajectories asymptotically reach their optimal values. Furthermore, we prove the forward invariance of the safe set by employing the notion of high-order barrier functions (HOBF), enforced through saddle point dynamics. Using modified high-gain observers and high-frequency perturbations, we present a control algorithm that estimates the unknown dynamics of the nonlinear system solely based on measurements of the cost function and constraints. Simulation results demonstrate the algorithm’s effectiveness in steering the system response towards its optimum while avoiding unsafe region.