Second-Order Newton-Based Extremum Seeking for Multivariable Dynamic Maps

A second-order Newton-based extremum seeking (SONES) algorithm is presented to estimate directional inflection points for multivariable dynamic maps. The design extends the first-order Newton-based extremum seeking algorithm that drives the system toward its peak point. This work provides perturbation matrices to estimate the second- and third-order derivatives necessary to implement the SONES. A set of conditions is provided for the probing frequencies that ensure accurate estimation of the derivatives. The algorithm utilizes high-order low-pass filters to improve its stability margin and transient performance. A differential Riccati filter calculates the inverse of the third-order derivative. The stability of the new algorithm is proved using the finite-horizon averaging theorem and singular perturbation analysis. The proposed algorithm ensures uniform convergence toward the directional inflection point without requiring information about the curvature of the map and its gradient. Simulation results show the effectiveness of the proposed algorithm for finding directional inflection points in dynamic maps and the advantages of high-order low-pass filters on the overall performance of the SONES.