Gradient- and Newton-Based Unit Vector Extremum Seeking Control

This paper presents novel methods for achieving stable and efficient convergence in multivariable extremum seeking control (ESC) using sliding mode techniques. Drawing inspiration from both classical sliding mode control and more recent developments in finite-time and fixed-time control, we propose a new framework that integrates these concepts into Gradient- and Newton-based ESC schemes based on sinusoidal perturbation signals. The key innovation lies in the use of discontinuous "relay-type" control components, replacing traditional proportional feedback to estimate the gradient of unknown quadratic nonlinear performance maps with Unit Vector Control (UVC). This represents the first attempt to address real-time, model-free optimization using sliding modes within the classical extremum seeking paradigm. In the Gradient-based approach, the convergence rate is influenced by the unknown Hessian of the objective function. In contrast, the Newton-based method overcomes this limitation by employing a dynamic estimator for the inverse of the Hessian, implemented via a Riccati equation filter. We establish finite-time convergence of the closed-loop average system to the extremum point for both methods by leveraging Lyapunov-based analysis and averaging theory tailored to systems with discontinuous right-hand sides. Numerical simulations validate the proposed method, illustrating significantly faster convergence and improved robustness compared to conventional ESC strategies, which typically guarantee only exponential stability. The results also demonstrate that the Gradient-based method exhibits slower convergence and higher transients since the gradient trajectory follows the curved and steepest-descent path, whereas the Newton-based method achieves faster convergence and improved overall performance going straightly to the extremum.