Logarithmic Barrier Functions for Practically Safe Extremum Seeking Control

Abstract

This paper presents a methodology for Practically Safe Extremum Seeking (PSfES), designed to optimize unknown objective functions while strictly enforcing safety constraints via a Logarithmic Barrier Function (LBF). Unlike traditional safety-filtered approaches that may induce chattering, the proposed method augments the cost function with an LBF, creating a repulsive potential that penalizes proximity to the safety boundary. We employ averaging theory to analyze the closed-loop dynamics. A key contribution of this work is the rigorous proof of practical safety for the original system. We establish that the system trajectories remain confined within a safety margin, ensuring forward invariance of the safe set for a sufficiently fast dither signal. Furthermore, our stability analysis shows that the model-free ESC achieves local practical convergence to the modified minimizer strictly within the safe set, through the sequential tuning of small parameters. The theoretical results are validated through numerical simulations.

Figures (4)

• Figure B1: Block diagram of the multi-dimensional combined ESC and LBF.
• Figure E1: Simulation in the 1D case. The initial position of the seeker is ${\theta_{\text{ESC}} = \theta_{\text{LBF}} = \theta_{\text{CBF}} = -3}$, and the unconstrained optimum is at $\theta^* = 0$. The controller parameters are set as follows: $a = 0.25$, $k = 0.2$, $\omega = 15$, and $\mu = 3$ for the LBF-ESC; and $a = 0.25$, $k = 0.3$, $\omega = 15$, $\omega_h=\omega_\ell=4.5$, $c = 0.1$, and $\delta = 0.001$ for the CBF-ESC.
• Figure E2: Simulation of a 2D point mass case with island-shaped obstacle avoidance. The initial position of the seeker is $(\theta_1, \theta_2) = (0, -4)$, and the unconstrained optimum is at $\theta^* = (4, 4)$. The controller parameters are set as follows: $a = 0.25$, $k = 0.01$, $\omega_1 = 75$, $\omega_2 = 100$, and $\mu = 6$ for the LBF-ESC; and $a = 0.25$, $k = 0.1$, $\omega_1 = 75$, $\omega_2 = 100$, $\omega_h=\omega_\ell=30$, $c = 0.5$, and $\delta = 0.001$ for the CBF-ESC.
• Figure E3: Simulation of a 2D point mass case with passage navigation. The obstacles are located at $(-3, 1)$ and $(1, 3)$ with radii equal to $2$ and $1.5$, respectively. The initial position of the seeker is $(\theta_1, \theta_2) = (0, -4)$, and the unconstrained optimum is at $\theta^* = (-3, 4)$. The controller parameters are set as follows: $a = 0.25$, $k = 0.01$, $\omega_1 = 75$, $\omega_2 = 100$, and $\mu = 6$ for the LBF-ESC; and $a = 0.25$, $k = 0.1$, $\omega_1 = 75$, $\omega_2 = 100$, $\omega_h=\omega_\ell=30$, $c = 1$, and $\delta = 0.001$ for the CBF-ESC.

Theorems & Definitions (4)

• Theorem 1: Practical Safety via Forward Invariance
• Definition 1: modified from ref:todorovski-2025
• Lemma 1
• Theorem 2