Exponential Extremum Seeking with Unbiased Convergence
Cemal Tugrul Yilmaz, Mamadou Diagne, Miroslav Krstić
Abstract
We present a multivariable extremum seeking (ES) algorithm for static and dynamic maps that achieves unbiased convergence to the optimum exponentially, referred to as exponential ES. The conventional ES approach, which uses constant amplitude sinusoids, results in steady-state oscillations around the optimum and is unable to guarantee unbiased convergence. In contrast, our ES approach employs exponential decay and growth functions to gradually decrease the amplitude of the perturbation signal and increase the amplitude of the demodulation signal, respectively. This eliminates the steady-state oscillation. To achieve unbiased convergence, we choose an adaptation gain that is sufficiently larger than the decay rate of the perturbation so that the learning process outpaces the perturbation's waning. The stability analysis is based on state transformation, averaging, and singular perturbation methods applied to the transformed system resulting in local stability of the transformed system as well as local exponential stability of the original system. For numerical simulation, we consider the problem of source seeking by a 2D velocity actuated point.