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

Sameer Pokhrel, Sameh A. Eisa|arXiv (Cornell University)|Oct 11, 2023

Extremum Seeking Control Systems4 citations

TL;DR

This paper establishes a rigorous connection between higher-order Lie bracket approximations and higher-order averaging in control-affine systems using chronological calculus, proving that the n-th order Lie bracket approximation corresponds to the (n+1)-th order averaging term. The framework eliminates unproven assumptions from prior work and enables faster convergence in extremum seeking by revealing higher-order dynamics for performance optimization.

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 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 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.

Motivation & Objective

• To provide a rigorous derivation of Lie bracket approximations in control-affine systems using chronological calculus.
• To bridge the gap between averaging theory and approximation theory in control-affine systems by showing Lie bracket approximations are higher-order averaging terms.
• To eliminate unproven assumptions in prior literature by deriving them as natural corollaries from the proposed framework.
• To apply the results to extremum seeking systems for improved convergence speed and performance using higher-order dynamics.
• To validate the theoretical findings with numerical simulations comparing against existing literature.

Proposed method

• Utilizes chronological calculus to systematically derive higher-order Lie bracket approximations from time-varying control-affine systems.
• Derives the n-th order Lie bracket approximation as equivalent to the (n+1)-th order averaging term, establishing a formal hierarchy.
• Applies the framework to control-affine extremum seeking systems by expressing the dynamics in terms of iterated Lie brackets and time-averaged coefficients.
• Computes higher-order averaging terms (up to fourth order) through recursive bracketing and integration over the period, yielding explicit expressions for the averaged vector fields.
• Transforms the results from the fast time scale to the slow time scale using a scaling factor involving ω^(p*-1), enabling practical application.
• Derives explicit formulas for the fourth-order averaged system, including coefficients involving integrals of trigonometric products and control input modulations.

Experimental results

Research questions

• RQ1 How can Lie bracket approximations in control-affine systems be formally linked to higher-order averaging theory?
• RQ2 What are the precise mathematical conditions under which Lie bracket approximations emerge as higher-order averaging terms?
• RQ3 Can the unproven assumptions in prior literature on Lie bracket approximations be derived as corollaries from a more general framework?
• RQ4 How can higher-order averaging terms improve the convergence rate in extremum seeking control systems?
• RQ5 What is the structure of the fourth-order averaged system in terms of iterated Lie brackets and control input modulation?

Key findings

• The n-th order Lie bracket approximation is rigorously shown to be equivalent to the (n+1)-th order averaging term in control-affine systems.
• The framework eliminates ad hoc assumptions from prior works by deriving them as natural consequences of chronological calculus.
• The fourth-order averaged system is explicitly derived, with terms involving double and triple Lie brackets and coefficients depending on ω^(p_i + p_j + p_k + p_l - 3).
• The derived fourth-order averaging term includes contributions from [ [b_i, b_j], [b_k, b_l] ], [ [[b_i, b_j], b_k], b_l ], and [ b_l, [[b_i, b_j], b_k] ] with time-averaged coefficients.
• The results enable faster convergence in extremum seeking by revealing higher-order dynamics that can be exploited in controller design.
• Numerical simulations demonstrate the significance of higher-order terms and validate improved performance over classical first- and second-order approximations.