Extremum Seeking with High-Order Lie Bracket Approximations: Achieving Exponential Decay Rate
Victoria Grushkovskaya, Sameh A. Eisa
TL;DR
The paper advances extremum seeking by leveraging higher-order Lie bracket averaging to achieve exponential-like convergence for cost functions with polynomial-like local behavior near a minimizer. By designing ES dynamics that excite high-order Lie brackets and using Chen-Fliess series, it enables trajectory motion along higher-order derivative directions of the cost, potentially outperforming traditional ES when $J(x)-J^* \\sim \\|x-x^*\\|^{m}$ with $m>2$. The main results show semi-global practical exponential stability for two-input ES under suitable bracket realizations and extend to multi-input designs and unknown degrees through dithering strategies that combine multiple orders. Numerical simulations on $J(x)=(x-1)^4$ corroborate faster convergence relative to classic ES, while highlighting oscillations that motivate further damping and gain-scheduling techniques. Overall, the work opens a promising direction for high-order Lie bracket methods in concrete ES implementations with improved convergence properties.
Abstract
This paper focuses on the further development of the Lie bracket approximation approach for extremum seeking systems. Classical results in this area provide extremum seeking algorithms with exponential convergence rates for quadratic-like cost functions, and polynomial decay rates for cost functions of higher degrees. This paper proposes a novel control design approach that ensures the motion of the extremum seeking system along directions associated with higher-order Lie brackets, thereby ensuring exponential convergence for cost functions that are polynomial-like but with degree greater than two.