Extremum Seeking Based on Approximated Sign of Gradient of Unknown Plant Maps

This paper presents a novel perturbation-based extremum seeking (ES) based on the smooth approximation of the sign of the gradient of unknown plant maps, which will improve the stability and the control performance of ES. The stability and the convergence speed of conventional sinusoidally perturbed gradient-based ES are very sensitive to the curvature of an unknown plant map, which has motivated the development of improved ES schemes that are less sensitive to the curvature such as Newton-like ES schemes. In this paper, an alternative ES scheme is proposed by estimating the sign of the gradient instead of the gradient itself. The ES scheme not only frees us from taking care of the curvature of the unknown map but also is able to minimize the increase of implementation complexity by just incorporating the signum function for the conventional ES scheme. Moreover, the signum function is replaced by its smooth approximation such as the softsign or a sigmoid function in order not only to suppress chattering phenomena but also to be able to adjust the behavior in the vicinity of an extremum point. It is also shown that the local-stability of the proposed ES scheme can be proved under some mild assumptions for the approximation function. The effectiveness of the proposed ES is demonstrated by some simulation examples for a simple process model and for a complex nonlinear biological wastewater treatment process model.