Extremum Seeking Feedback Under Unknown Hessian Signs

Extremum seeking control aims at determining and keeping the output of a nonlinear static or dynamic map on its unknown extremum point. In the literature, despite considering an unknown objective function in the real-time optimization problem, it is mandatory to know whether the extremum point is a maximum or a minimum, which is characterized by means of the Hessian sign of the nonlinear scalar map. This article proposes that the process of extremum seeking occurs independently of the Hessian sign information. The key idea is to combine the classical extremum seeking approach with a switching monitoring function-based scheme. The switching algorithm will drive the closed-loop system to the unknown extremum, neglecting if it is a maximum or a minimum equilibrium point. Theoretical results are rigorously justified and simulation examples show the robustness properties of the proposed recipe even when the changes of the Hessian sign occur in “on-the-fly” fashion, showing the adaptability of the proposed method to solve distinct online minimizing and maximizing problems in sequence.