  This paper provides a rigorous derivation for what is known in the literature
as the Lie bracket approximation of control-affine systems in a sequential
framework for higher-orders. In fact, by using chronological calculus, we show
that said Lie bracket approximations are themselves higher-order averaging
terms. Hence, this paper bridges both averaging and approximation theories of
control-affine systems. In particular, the Lie bracket approximation of order
($n$) turns out to be a higher-order averaging of order ($n+1$). The derivation
and formulation provided in this paper can be directly reduced to the first and
second-order Lie bracket approximations available in the literature. However,
we do not need to make many of the unproven assumptions provided in the
literature and show that they are in fact natural corollaries from our work.
Moreover, we use our results to show that important and useful information
about control-affine extremum seeking systems can be obtained and used for
performance improvement, including faster convergence of very simple
structures. We provide multiple numerical simulations to demonstrate both the
conceptual elements of this work as well as the significance of our results on
extremum seeking with comparison against the literature.
