Higher-order Lie bracket approximation and averaging of control-affine systems with application to extremum seeking

TL;DR

This paper unifies averaging and approximation theories for control-affine systems using higher-order Lie brackets and demonstrates improved extremum seeking performance through higher-order analysis.

Contribution

It introduces a general, rigorous framework for higher-order Lie bracket approximations using chronological calculus, extending existing theories without many previous assumptions.

Findings

Higher-order Lie bracket approximations are equivalent to higher-order averaging.

The framework reduces to known first and second-order cases.

Numerical simulations show improved extremum seeking convergence rates.

Abstract

This paper provides a rigorous derivation for what is known in the literature as the Lie bracket approximation of control-affine systems in a more general and sequential framework for higher-orders. In fact, by using chronological calculus, we show that said Lie bracket approximations can be derived, and considered, as higher-order averaging terms. Hence, the theory provided in this paper unifies 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 assumptions that were needed/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 significant performance improvement, including a faster convergence rate influenced by higher-order derivatives. 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.