Adaptive Extremum Seeking Controller Design Utilizing A Simple Nonlinear Benchmark Process Model

This paper presents an adaptive extremum seeking (ES) controller design method by employing a simple nonlinear benchmark process. A first-order plus quadratic nonlinear function model is chosen as the benchmark process model and a classical third-order sinusoidally perturbed ES scheme is considered. After deriving the average model of the ES scheme for the benchmark process, a guideline for the controller parameter tuning to shape the transient response of the average model is presented under the condition that the second derivative (Hessian) of the output is known. Then, a design method utilizing a second derivative estimator is presented in order to meet realistic situation where the second derivative is unknown and often time-varying. It is pointed out that the obtained ES scheme is essentially equivalent to a Newton-like ES scheme. It is also shown that the prerequsite knowledge about the process for the ES controller design is the process time constant only. Finally, simulation examples illustrate the effectiveness of the proposed ES design method by applying it not only for the benchmark process model but also for a complex nonlinear biological wastewater treatment process model.