Prescribed-Time Newton Extremum Seeking using Delays and Time-Periodic Gains

TL;DR

$PT$-ES tackles finite-in-time optimization for static maps with delays by introducing a Newton-like scheme that leverages time-periodic delayed feedback, bounded gains, and averaging over fast oscillations. The core method (KHV PT-ES) uses three filters (gradient, Hessian, and Riccati-like inverse-Hessian estimators) driven by oscillators to realize prescribed-time convergence to the optimizer while compensating map delays. A rigorous averaging-based analysis in Retarded Functional Differential Equations shows the averaged error dynamics converge in a prescribed time, yielding local guarantees with explicit dependence on the prescribed time $T^*$ and delays $D$, and a corollary extends to delay-free, multivariable settings. Numerical simulations validate the scalar-delayed and multivariable-delay-free cases, demonstrating fast convergence to the unknown optimizer and robust Hessian estimation within the prescribed horizon, suggesting practical applicability to fast autonomous optimization under measurement delays.

Abstract

We study prescribed-time extremum seeking (PT-ES) for scalar maps in the presence of time delays. The PT-ES problem has been studied by Yilmaz and Krstic in 2023 using chirpy probing and time-varying gains that grow unbounded. To alleviate the gain singularity, in this paper we present an alternative approach, employing delays with bounded time-periodic gains, for achieving prescribed-time convergence to the extremum. Our results are not extensions or refinements of earlier works, but a new methodological direction --applicable even when the map has no delay. The main PT-ES algorithm compensates the map's delay and uses perturbation-based and the Newton (rather than gradient) approaches. With the help of averaging theorems in infinite dimension, specifically Retarded Functional Differential Equations (RFDEs), we conduct a prescribed-time convergence analysis on a suitable averaged target ES system, which contains the time-periodic gains of the map and feedback delays. We further extend our method to multivariable static maps and illustrate our results through numerical simulations.

Figures (9)

• Figure 1: Left: Solution of \ref{['eq:single_integrator_in_coreIDEA']} with the $2 T$ - periodic time-varying pointwise delayed feedback \ref{['eq:time-varying_delayed-feedback_key_result_in_coreIDEA']}-\ref{['eq:mathcalK_T_time_sequence_in_coreIDEA']} with $T=1s$. Right: Profile of the $2 T$- periodic time-varying pointwise delayed feedback \ref{['eq:time-varying_delayed-feedback_key_result_in_coreIDEA']}-\ref{['eq:mathcalK_T_time_sequence_in_coreIDEA']}.
• Figure 2: Left: Solution of \ref{['eq:single_integrator_in_coreIDEA']} with the $2 T$ - periodic time-varying pointwise delayed feedback \ref{['eq:time-varying_delayed-feedback_key_result_in_coreIDEA']}-\ref{['eq:mathcalK_T_time_sequence_in_coreIDEA']} with $T=1s$, and which is subject to an additive vanishing input e.g., $v(t)=1$ for $t < 1s$, and $v(t)=0$ for $t \geq 1s$. Right: Profile of the $2 T$- periodic time-varying pointwise delayed feedback \ref{['eq:time-varying_delayed-feedback_key_result_in_coreIDEA']}-\ref{['eq:mathcalK_T_time_sequence_in_coreIDEA']}.
• Figure 3: Left: Profile of the $2T$- periodic signal $\mathcal{K}_T(t)$ defined in \ref{['eq:mathcalK_T']} generated by the linear constrained oscillator \ref{['eq:oscillator_system_periodic_time-varying_feedbacks']} with $T=1s$ (blue line) and $T=2s$ (blue dashed line), and initial condition $\zeta(0)=(0,1)^{\top}\in \mathbb{S}^1$, which gives $\delta_{\zeta} = 0$, according to \ref{['eq:definition_of_delta-phase_of_oscillators']}. Right: Profile of the $2T$- periodic signal $\mathcal{L}_T(t)$ defined in \ref{['eq:mathcalL_T']} generated by the linear constrained oscillator \ref{['eq:oscillator_system_periodic_time-varying_feedbacks']} with $T=1s$ (red line) and $T=2s$ (red dashed line), and initial condition $\zeta(0)=(0,1)^{\top}\in \mathbb{S}^1$, which gives $\delta_{\zeta} = 0$, according to \ref{['eq:definition_of_delta-phase_of_oscillators']}.
• Figure 4: Scheme of the KHV PT-ES \ref{['eq:Newton_ES_static_maps1_prescribed-pointwise']}-\ref{['eq:Newton_ES_static_maps3_prescribed-pointwise_Hessian']} with map delay $D$ and feedback delays $T$ and $2T$.
• Figure 5: Details of the three "filters" in Figure \ref{['PT_ES_diagram']}. Notation is abused copiously (simultaneous time-domain and Laplace-domain nomenclature) to convey intuition. Top: The $z$-filter, given in \ref{['eq:Newton_ES_static_maps2_prescribed-pointwise']}, mirrors the structure of the exponential filter \ref{['eq:Newton_ES_static_maps2V2']}. The $z$-filter is fed by $M(t)y(t)$, an estimate of the "scalar gradient" $Q'(\theta(t-D))$. This estimate drives a predictor for the approximation of a "Newton update" $\Gamma Q'(\theta(t-D))$, cascaded into a KHV-type low-pass filter (prescribed time, employing time-periodic gains and delay feedback). The approximation $\Gamma(t) M(t) y(t) \approx \Gamma(t) Q'(\theta(t-D))$ is the product of $\Gamma$, an estimate of the inverse Hessian $1/H^{\star}$, and $My$, an approximation of the scalar gradient $Q'(\hat{\theta}(t-D)) = H^{\star} \tilde{\theta}(t-D)$, as in \ref{['eq:Newton_ES_static_maps2V2']}. The KHV filter plays a role analogous to the exponential filter $c/(s+c)$ in \ref{['eq:Newton_ES_static_maps2_prescribed-pointwise']}. Although not needed in practice--neither in \ref{['eq:Newton_ES_static_maps2V2']} nor in \ref{['eq:Newton_ES_static_maps2_prescribed-pointwise']}--this low-pass filtering enables the use of the averaging theorem in Hale1990, as the closed-loop system without it does not reduce to a standard RFDE. Middle: The $\hat{H}$-filter, given in \ref{['eq:Newton_ES_static_maps3_prescribed-pointwise_Hessian']}, is a low-pass filter of the KHV kind applied to the Hessian estimate $N(t)y(t)$. Bottom: The $\Gamma$-filter in \ref{['eq:Newton_ES_static_maps3_prescribed-pointwise_Riccati']} is a KHV version of the exponential Riccati-type filter in \ref{['eq:Newton_ES_static_maps3V2']}, estimating in prescribed time the inverse Hessian $1/H^{\star}$. Since $\hat{H}$ approximates $H^{\star}$ and $\Gamma$ approximates $1/H^{\star}$, the product $\Gamma(t) \hat{H}(t - 2T)$ should be regarded as $\approx 1$, in the sense of design inspiration.

Theorems & Definitions (13)

• Remark 1
• Remark 2
• Remark 3
• Lemma 1
• proof
• Remark 4
• Lemma 2
• Claim 1
• Theorem 1
• Corollary 1