  Control-affine Extremum Seeking Control (ESC) systems have been increasingly
studied and applied in the last decade. In a recent effort, many control-affine
ESC structures have been generalized in a unifying class and their stability
was analyzed. However, guaranteeing vanishing oscillations at the extremum
point for said class requires strong conditions that may not be feasible or
easy to check/design by the user, especially when the gradient of the objective
function is unknown. In this paper, we introduce a control-affine ESC structure
that remedies this problem such that: (i) its oscillations attenuate
structurally via a novel application of a geometric-based Kalman filter and a
Lie bracket estimation approach; and (ii) its stability is characterized by a
time-dependent (one-bound) condition that is easier to check and relaxed when
compared to the generalized approach mentioned earlier. We provide numerical
simulations of three problems to demonstrate the effectiveness of our proposed
ESC; these problems cannot be solved with vanishing oscillations using the
generalized approach in the literature.
