[Paper Review] Fixed-Time Extremum Seeking

Jorge I. Poveda, Miroslav Krstić|arXiv (Cornell University)|Dec 15, 2019

Extremum Seeking Control Systems41 references4 citations

TL;DR

This paper introduces fixed-time extremum seeking controllers that guarantee convergence to the optimal solution within a prescribed, initial-condition-independent time bound. Using novel gradient-based and Newton-based dynamics derived via averaging theory for non-smooth systems, the proposed algorithms achieve fixed-time convergence for static and dynamical systems, with the Newton-based variant offering fully assignable convergence times independent of the cost function's properties.

ABSTRACT

We introduce a new class of extremum seeking controllers able to achieve fixed time convergence to the solution of optimization problems defined by static and dynamical systems. Unlike existing approaches in the literature, the convergence time of the proposed algorithms does not depend on the initial conditions and it can be prescribed a priori by tuning the parameters of the controller. Specifically, our first contribution is a novel gradient-based extremum seeking algorithm for cost functions that satisfy the Polyak-Lojasiewicz (PL) inequality with some coefficient κ> 0, and for which the extremum seeking controller guarantees a fixed upper bound on the convergence time that is independent of the initial conditions but dependent on the coefficient κ. Second, in order to remove the dependence on κ, we introduce a novel Newton-based extremum seeking algorithm that guarantees a fully assignable fixed upper bound on the convergence time, thus paralleling existing asymptotic results in Newton-based extremum seeking where the rate of convergence is fully assignable. Finally, we study the problem of optimizing dynamical systems, where the cost function corresponds to the steady-state input-to-output map of a stable but unknown dynamical system. In this case, after a time scale transformation is performed, the proposed extremum seeking controllers achieve the same fixed upper bound on the convergence time as in the static case. Our results exploit recent gradient flow structures proposed by Garg and Panagou in [3], and are established by using averaging theory and singular perturbation theory for dynamical systems that are not necessarily Lipschitz continuous. We confirm the validity of our results via numerical simulations that illustrate the key advantages of the extremum seeking controllers presented in this paper.

Motivation & Objective

• To address the persistent challenge of slow transient performance in extremum seeking control, particularly the unbounded convergence time that grows with initial conditions.
• To develop model-free extremum seeking algorithms that achieve fixed-time convergence—i.e., convergence within a prescribed upper bound independent of initial conditions—without requiring knowledge of the cost function's derivatives.
• To extend fixed-time convergence guarantees to dynamical systems by leveraging time-scale transformations and steady-state input-to-output mappings.
• To establish theoretical foundations using generalized averaging and singular perturbation theory for non-Lipschitz systems, enabling robust convergence analysis.
• To provide a framework for future extensions to constrained optimization, game-theoretic settings, and time-varying optimizers with fixed-time convergence.

Proposed method

• Proposes a fixed-time gradient-based extremum seeking (FTGES) algorithm for cost functions satisfying the Polyak-Lojasiewicz (PL) inequality, with convergence time upper bound dependent on the PL constant κ.
• Introduces a fixed-time Newton-based extremum seeking (FTNES) algorithm that removes dependence on κ, enabling fully assignable convergence time bounds via controller parameters.
• Employs a time-scale separation structure with high-frequency oscillations and slow dynamics, where the slow dynamics emulate gradient or Newton flows via averaging theory.
• Applies generalized averaging theory to non-smooth, non-Lipschitz systems to analyze convergence, extending classical results to non-smooth extremum seeking.
• Performs time-scale transformations on dynamical systems to map their steady-state input-to-output behavior into a form where fixed-time convergence can be guaranteed.
• Derives reduced average dynamics that match standard gradient or Newton flows, ensuring the closed-loop system tracks the optimal solution within a fixed time.

Experimental results

Research questions

• RQ1 Can extremum seeking controllers be designed to achieve fixed-time convergence independent of initial conditions for static optimization problems?
• RQ2 Can the convergence time bound be made fully assignable and independent of the cost function's properties, particularly the PL constant κ?
• RQ3 Can fixed-time convergence be extended to dynamical systems with unknown stable dynamics by leveraging steady-state input-to-output mappings?
• RQ4 Can averaging theory be generalized to analyze non-smooth, non-Lipschitz systems arising in extremum seeking?
• RQ5 Can the proposed framework be extended to more complex settings such as constrained optimization or time-varying optimizers with fixed-time convergence?

Key findings

• The FTGES algorithm achieves fixed-time convergence with an upper bound on convergence time that depends on the PL constant κ, but is independent of initial conditions.
• The FTNES algorithm achieves a fully assignable fixed upper bound on convergence time, independent of κ and initial conditions, by leveraging Newton-like dynamics.
• For dynamical systems, after a time-scale transformation, the extremum seeking controllers achieve the same fixed upper bound on convergence time as in the static case.
• The reduced average dynamics of the proposed controllers match standard gradient or Newton flows, ensuring convergence to the global minimizer under appropriate assumptions.
• Numerical simulations confirm the theoretical results, demonstrating fast, bounded convergence independent of initial conditions.
• The theoretical analysis is established using generalized averaging and singular perturbation theory for non-Lipschitz systems, extending prior results to non-smooth extremum seeking.