[Paper Review] A Robust Time-Delay Approach to Extremum Seeking via ISS Analysis of the Averaged System

Xuefei Yang, Emilia Fridman|arXiv (Cornell University)|Feb 3, 2023

Extremum Seeking Control Systems4 citations

TL;DR

This paper presents a robust time-delay approach to extremum seeking for N-dimensional static quadratic maps in both continuous and discrete domains by transforming the system into a neutral-type time-delay model. Using variation of constants formula instead of Lyapunov-Krasovskii methods, it establishes explicit, less conservative conditions for practical stability via input-to-state stability (ISS) analysis, enabling larger Hessian uncertainty bounds and more lenient dither period constraints than prior work.

ABSTRACT

For N-dimensional (ND) static quadratic map, we present a time-delay approach to gradient-based extremum seeking (ES) both, in the continuous and, for the first time, the discrete domains. As in the recently introduced (for 2D maps in the continuous domain), we transform the system to the time-delay one (neutral type system in the form of Hale in the continuous case). This system is O($\varepsilon$)-perturbation of the averaged linear ODE system, where $\varepsilon$ is a period of averaging. We further explicitly present the neutral system as the linear ODE, where O($\varepsilon$)-terms are considered as disturbances with distributed delays of the length of the small parameter $\varepsilon$. Regional input-to-state stability (ISS) analysis is provided by employing a variation of constants formula that greatly simplifies the previously used analysis via Lyapunov-Krasovskii (L-K) method, simplifies the conditions and improves the results. Examples from the literature illustrate the efficiency of the new approach, allowing essentially large uncertainty of the Hessian matrix with bounds on $\varepsilon$ that are not too small.

Motivation & Objective

• To develop a robust, quantitative stability analysis framework for extremum seeking (ES) in multi-variable systems with arbitrary N ≥ 1.
• To overcome the limitations of Lyapunov-Krasovskii (L-K) methods, which yield conservative results and complex conditions under Hessian uncertainty.
• To extend the time-delay approach from 1–2D systems to general N-dimensional systems in both continuous and discrete domains.
• To provide explicit, simple inequality-based conditions for practical stability, enabling larger allowable dither periods and Hessian uncertainties.

Proposed method

• Transform the gradient-based extremum seeking dynamics into a neutral-type time-delay system, analogous to Hale's formulation in the continuous case.
• Reformulate the system as a linear ordinary differential equation (ODE) perturbed by O(ε)-terms with distributed delays of length ε, where ε is the averaging period.
• Apply the variation of constants formula to analyze input-to-state stability (ISS) of the averaged system, replacing the complex L-K functional approach.
• Derive explicit, simple inequalities to bound the dither period ε and ensure practical stability, avoiding the conservatism of L-K methods.
• Use the ISS framework to quantify the system's robustness to disturbances from Hessian uncertainty and initial condition errors.
• Validate the approach through examples from the literature, demonstrating improved stability margins under large Hessian uncertainty.

Experimental results

Research questions

• RQ1 Can a time-delay approach be generalized to N-dimensional extremum seeking systems in both continuous and discrete domains?
• RQ2 How can the complexity and conservatism of Lyapunov-Krasovskii methods be reduced in stability analysis of extremum seeking systems?
• RQ3 What explicit, simple conditions can guarantee practical stability of extremum seeking systems under large Hessian uncertainty?
• RQ4 Can the dither period ε be allowed to be larger while preserving stability, compared to existing L-K-based methods?
• RQ5 To what extent does the variation of constants formula improve the tractability and tightness of stability bounds in extremum seeking?

Key findings

• The proposed method achieves practical stability for N-dimensional extremum seeking systems using a time-delay model that is O(ε)-perturbed by distributed delays.
• Explicit stability conditions are derived as simple inequalities, avoiding the need for complex L-K functional construction and reducing conservatism.
• The method allows for significantly larger bounds on Hessian matrix uncertainty compared to prior L-K-based approaches.
• The upper bound on the dither period ε is less restrictive, enabling faster convergence and improved robustness in practical implementations.
• The variation of constants formula simplifies stability analysis and leads to tighter, more practical stability margins than previous methods.
• Numerical examples confirm that the method maintains stability even when ε is not very small, demonstrating its practical viability.