Extremum Seeking for Single-Variable Time-Varying Objective Functions

This paper addresses the problem of tracking time-varying optimal trajectories for convex optimization problems where the objective function is time-varying and its explicit form is unknown but measurable. We propose a discrete-time extremum seeking algorithm that leverages sampled-data measurements to approximate derivatives and iteratively update the system state. The algorithm employs five sampling points within each interval to estimate the gradient, Hessian, and mixed time derivatives of the objective function via finite difference approximations. Under the assumptions of uniform strong convexity and smoothness of the objective function with bounded derivatives, we establish the practical stability of the algorithm and derive an ultimate bound for the tracking error. The key innovation lies in transforming the continuous-time dynamic optimization problem into a discrete iterative framework while rigorously quantifying approximation errors. The efficiency of the new approach is demonstrated by an example.