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Second-Order Newton-Based Extremum Seeking for Multivariable Static Maps

Azad Ghaffari, Tiago Roux Oliveira

TL;DR

The paper tackles real-time estimation of directional inflection points in multivariable static maps by introducing a second-order Newton-based ES (SONES). It develops perturbation-based schemes to estimate the second- and third-order derivatives of $y=h(\theta)$, utilizing specially designed matrices $N(t)$ and $P(t)$ and averaging to isolate higher-order terms, with the inverse of the third-order derivative computed via a differential Riccati filter. Under averaging theory, the authors prove local exponential stability of the averaged SONES dynamics for small perturbations and provide conditions on probing frequencies to ensure accurate derivative estimates. A simulation on a crafted map demonstrates convergence to the directional inflection point along a chosen axis, confirming robust performance near saddle and local extrema. The work offers a viable, model-free framework for multivariable extremum seeking with potential extensions to dynamic maps and broader applications in real-time optimization.

Abstract

A second-order Newton-based extremum seeking (SONES) algorithm is presented to estimate directional inflection points for multivariable static maps. The design extends the first-order Newton-based extremum seeking algorithm that drives the system toward its peak point. This work provides perturbation matrices to estimate the second- and third-order derivatives necessary for implementation of the SONES. A set of conditions are provided for the probing frequencies that ensure accurate estimation of the derivatives. A differential Riccati filter is used to calculate the inverse of the third-order derivative. The local stability of the new algorithm is proven for general multivariable static maps using averaging analysis. The proposed algorithm ensures uniform convergence toward directional inflection point without requiring information about the curvature of the map and its gradient. Simulation results show the effectiveness of the proposed algorithm.

Second-Order Newton-Based Extremum Seeking for Multivariable Static Maps

TL;DR

The paper tackles real-time estimation of directional inflection points in multivariable static maps by introducing a second-order Newton-based ES (SONES). It develops perturbation-based schemes to estimate the second- and third-order derivatives of , utilizing specially designed matrices and and averaging to isolate higher-order terms, with the inverse of the third-order derivative computed via a differential Riccati filter. Under averaging theory, the authors prove local exponential stability of the averaged SONES dynamics for small perturbations and provide conditions on probing frequencies to ensure accurate derivative estimates. A simulation on a crafted map demonstrates convergence to the directional inflection point along a chosen axis, confirming robust performance near saddle and local extrema. The work offers a viable, model-free framework for multivariable extremum seeking with potential extensions to dynamic maps and broader applications in real-time optimization.

Abstract

A second-order Newton-based extremum seeking (SONES) algorithm is presented to estimate directional inflection points for multivariable static maps. The design extends the first-order Newton-based extremum seeking algorithm that drives the system toward its peak point. This work provides perturbation matrices to estimate the second- and third-order derivatives necessary for implementation of the SONES. A set of conditions are provided for the probing frequencies that ensure accurate estimation of the derivatives. A differential Riccati filter is used to calculate the inverse of the third-order derivative. The local stability of the new algorithm is proven for general multivariable static maps using averaging analysis. The proposed algorithm ensures uniform convergence toward directional inflection point without requiring information about the curvature of the map and its gradient. Simulation results show the effectiveness of the proposed algorithm.
Paper Structure (6 sections, 1 theorem, 31 equations, 5 figures)

This paper contains 6 sections, 1 theorem, 31 equations, 5 figures.

Table of Contents

  1. Introduction
  2. High-Order Derivatives of Static Map
  3. Second-Order Newton-Based ES
  4. Simulation Results
  5. Conclusions
  6. Conditions on Probing Frequencies

Key Result

Theorem 1

Consider system eq:sysavg under Assumption assumption1. There exist $\bar{\delta}, \bar{a}>0$ such that for all $\delta\in(0,\bar{\delta})$ and $|a|\in(0,\bar{a})$ system eq:sysavg has a unique exponentially stable periodic solution $(\tilde{\theta}^{{\Pi}}(t),\hat{H}_m^{{\Pi}}(t),\tilde{\Lambda}_m^ for all $t\ge0$, where for all $i,j\in\!\!\{1, 2, \cdots, p\}$.

Figures (5)

  • Figure 1: Second-order gradient-based extremum seeking algorithm to find the directional inflection point along $\theta_m$-axis, where $K$ is a diagonal matrix with positive elements.
  • Figure 2: Second-order Newton-based ES proposed for estimating the inflection point along $\theta_m$-axis.
  • Figure 3: (a) The second-order Newton-based ES algorithm governs the system trajectory toward the inflection point along $\theta_1$-axis at $\theta^*=[1~~2]^\top$. The level sets of $y=h(\theta)$ indicate that the inflection point is near a local minimumu point and a saddle point at $[-0.07~~2.22]^\top$ and $[1.26~~ 2.62]^\top$, respectively. (b) The estimate of the parameters versus time.
  • Figure 4: (a) System trajectory among level sets of $G_1(\theta)=\partial h(\theta)/\partial\theta_1$. The inflection point along $\theta_1$-axis leads to a maximum point at $\theta^*$. (b) System trajectory among level sets of $G_2=\partial h(\theta)/\partial\theta_2$ indicating no extremum point.
  • Figure 5: Time evolution of the estimation of $\Lambda_1=T_1^{-1}$. The true value of $\Lambda_1$ is reached under $15$ seconds.

Theorems & Definitions (1)

  • Theorem 1