Speaker
Description
This talk introduces a new approach for solving extremum seeking problems. Extremum seeking methods are widely employed for real time optimization without requiring explicit knowledge of the cost function’s gradient. However, conventional approaches relying on constant gain strategies often lead to the practical asymptotic stability. This means that the trajectories of a system converge to a neighborhood of the optimum, with the radius of this neighborhood decreasing as the control amplitude and frequency increase. Our work overcomes these limitations by leveraging a time-varying gain, whose properties ensure improved convergence characteristics. Namely, we establish conditions on the gain under which the trajectories of the system asymptotically converge to the optimum. Key requirements include the monotonic decrease of the gain and certain integral bounds over time. This approach is inspired by conceptual similarities between stochastic gradient descent and extremum seeking algorithms. We demonstrate that the derived sufficient convergence conditions are conceptually similar to step-size rules in stochastic gradient descent and stochastic approximation algorithms. The theoretical foundation is based on Lie bracket approximations, however, unlike the classical framework, our approach does not require uniform asymptotic stability of the optimal state for the associated Lie bracket system. Additionally, we estimate the speed of convergence of the solutions to the derived extremum seeking system and investigate extremum seeking systems with time-varying frequencies. The obtained results are demonstrated through numerical simulations.
