On the accuracy of gradient estimation in extremum-seeking control using small perturbations

In many extremum-seeking control methods, perturbations are added to the parameter signals to estimate derivatives of the objective function (that is, the steady-state parameter-to-performance map) in order to optimize the steady-state performance of the plant using derivative-based algorithms. However, large perturbations are often undesirable or inapplicable due to practical constraints and a high cost of operation. Yet, many extremum-seeking control algorithms rely solely on perturbations to estimate all required derivatives. The corresponding derivative estimates, especially the Hessian and higher-order derivatives, may be qualitatively poor if the perturbations are small. In this work, we investigate the use of the nominal parameter signals in addition to the perturbations to improve the accuracy of the gradient estimate. In turn, a more accurate gradient estimate may result in a faster convergence and may allow for a higher tuning-gain selection. In addition, we show that, if accurate curvature information of the objective function is available via estimation or a priori knowledge, it may be used to further enhance the accuracy of the gradient estimate.