Multivariable Stochastic Newton-Based Extremum Seeking with Delays

TL;DR

The paper tackles real-time optimization of an unknown nonlinear map $y=Q(\theta)$ in a multi-input setting with distinct input delays. It introduces a Newton-based stochastic extremum-seeking controller that uses stochastic perturbations to estimate the Hessian inverse $H^{-1}$ via $\hat{H}(t)=N(\eta(t-D))y(t)$ and predictor feedback to compensate delays, with stability proved through backstepping and infinite-dimensional averaging. A transport-PDE delay representation, an averaged model, a Lyapunov-Krasovskii functional, and the stochastic averaging theorem together guarantee exponential convergence to a neighborhood of the extremum $\theta^*$ with a residue $\mathcal{O}(\epsilon)$ where $\epsilon=1/\omega$. Simulations under equal and distinct delays demonstrate effective delay compensation, rapid convergence relative to gradient-based ESC, and robustness of Hessian-adaptive updates, highlighting practical applicability to delay-sensitive, distributed-parameter systems.

Abstract

This paper presents a Newton-based stochastic extremum-seeking control method for real-time optimization in multi-input systems with distinct input delays. It combines predictor-based feedback and Hessian inverse estimation via stochastic perturbations to enable delay compensation with user-defined convergence rates. The method ensures exponential stability and convergence near the unknown extremum, even under long delays. It extends to multi-input, single-output systems with cross-coupled channels. Stability is analyzed using backstepping and infinite-dimensional averaging. Numerical simulations demonstrate its effectiveness in handling time-delayed channels, showcasing both the challenges and benefits of real-time optimization in distributed parameter settings.

Figures (13)

• Figure 1: Block diagram of the basic prediction scheme for compensating distinct input delays in stochastic extremum seeking using the Newton algorithm. The predictor feedback (\ref{['eq:29']}) is shown in its vector form with $F(s)=diag\{c_1/(s+c_1),...,c_n/(s+c_n)\}$ and $K=diag\{k_1,...,k_n\}$. The red block indicates the introduced delays, while the blue blocks show modifications to the classical stochastic Newton-based ESC algorithm LIUFinal for mitigating the effects of time delays.
• Figure 2: Evolution of the system input $\theta(t)$ under the multivariable Newton-based stochastic ESC when both input channels experience equal delays.
• Figure 3: Evolution of the system outputs $y(t)$ under the multivariable gradient-based and Newton-based stochastic ESC without delays.
• Figure 4: Evolution of the system outputs $y(t)$ for the multivariable gradient-based and Newton-based stochastic ESC with predictor feedback when input channels experience equal delays.
• Figure 5: Evolution of the system input $\theta(t)$ under the multivariable Newton-based stochastic ESC without delays.