Fast extremum seeking for a class of non Hammerstein-Wienner systems

This work addresses an extension of the classical design for perturbation-based extremum seeking control (ESC) method to a class of non Hammerstein-Wienner (HW) plants. The original ESC scheme, when applied to a non HW system, requires a very low perturbation frequency. This order of frequencies result in excessively larger settling times, rendering the ESC method impractical for certain applications. This work addresses this problem by presenting one approach named pre-compensated ESC (PC-ESC). The proposed technique approximates a class of higher-order non HW systems with unknown parameters to a Wienner model by probing the plant in order to estimate some unmeasured signal by means of a phase-locked loop (PLL) estimator. Such a signal brings necessary information of phase and gain of the perturbation signal in the output of the plant used in general for gradient reconstruction. The proposed ESC modification was numerically validated considering a class of plants that captures the essential dynamics of the Eikrem's model for gas lifted oil wells. The novel fast ESC applied to this context indicates that it is possible to maintain the oil production around the optimum point of the well-performance curve (WPC) in gas lifted oil wells, without reducing the frequency of the perturbation to impracticable values.