Characterizing the learning dynamics in extremum seeking
Provides a theoretical foundation for understanding and improving extremum seeking control, relevant for optimization of unknown functions in control systems.
The paper characterizes the learning dynamics of perturbation-based extremum seeking, showing it recovers an averaged gradient and optimizes a modified objective function, which can smooth out local extrema for global optimization.
We consider perturbation-based extremum seeking, which recovers an approximate gradient of an analytically unknown objective function through measurements. Using classical needle variation analysis, we are able to explicitly quantify the recovered gradient in the scalar case. We reveal that it corresponds to an averaged gradient of the objective function, even for very general extremum seeking systems. From this, we create a recursion which represents the learning dynamics along the recovered gradient. These results give rise to the interpretation that extremum seeking actually optimizes a function other than the original one. From this insight, a new perspective on global optimization of functions with local extrema emerges: because the gradient is averaged over a certain time period, local extrema might be evened out in the learning dynamics. Moreover, a multidimensional extension of the scalar results is given.