Gradient‐ and Newton‐Based Unit Vector Extremum Seeking Control

ABSTRACT This paper proposes new methods for stable and efficient convergence in multivariable extremum seeking control (ESC) using sliding mode techniques. Inspired by classical sliding modes and finite‐time control, the approach integrates these ideas into gradient‐ and Newton‐based ESC schemes with sinusoidal perturbations. The main innovation is the use of a discontinuous relay‐type control law based on Unit Vector Control (UVC) to estimate the gradient of unknown quadratic performance maps, replacing the traditional proportional feedback. This enables real‐time, model‐free optimization within the ESC framework. While the gradient‐based method exhibits a convergence rate that depends on the unknown Hessian, the Newton‐based approach overcomes this limitation by employing a Riccati‐based estimator of the inverse Hessian. Finite‐time convergence is established using Lyapunov analysis and averaging theory. Numerical simulations validate the proposed methods, illustrating significantly faster convergence and improved robustness of the UVC strategy compared with conventional ESC schemes, which typically guarantee only exponential stability.