Robustness and averaging properties of a large-amplitude, high-frequency extremum seeking control scheme

Abstract

We investigate robustness and averaging properties of an extremum seeking scheme which employs oscillatory dither signals with sufficiently large amplitudes and frequencies. Our study takes both input and output disturbances into account. We consider general L∞-disturbances, which may correlate with the oscillatory dither signals. A suitable change of variables followed by an averaging procedure reveals that the closed-loop system approximates the behavior of an averaged system. The disturbances in the averaged system are directly linked to the disturbances in the closed-loop system. This allows us to conclude stability and robustness properties of closed-loop system from respective properties of the averaged system. In particular, we show that, if the averaged system tolerates L∞-disturbances of a certain magnitude, then the same is true for the approximating closed-loop system. Another feature of the proposed method is that it steers the closed-loop system approximately into descent directions of an averaged objective function. This property can be beneficial for the purpose of global optimization because an averaged objective function is expected to have fewer undesired critical points than the original objective function.